#help-41 | Mathematics | Page 14

jovial field Sep 20, 2024, 5:36 PM

#

So

#

What should we do

patent raptor Sep 20, 2024, 5:37 PM

#

,, \sum_{n=1}^{\infty} \frac{[(2n)!]^2}{[2^nn!]^4} \approx \sum_{n=1}^{\infty} \frac{2\pi (2n) \left (\frac{2n}{e} \right )^{2n}}{2^{4n}(2\pi n)^2 \left (\frac{n}{e} \right )^{4n}} = \sum_{n=1}^\infty \frac{e^{2n}}{\pi \cdot 4^n \cdot n^{2n+1}}

jovial field Sep 20, 2024, 5:38 PM

#

Bro but I don't think I could use the approximation

#

See

#

Bro let's leave it for a biy

grizzled pagodaBOT Sep 20, 2024, 5:38 PM

#

bacc

jovial field Sep 20, 2024, 5:38 PM

#

Bit.

#

Let's get the other 16-17-18-19-20 done

patent raptor Sep 20, 2024, 5:39 PM

#

jovial field Bro but I don't think I could use the approximation

Why

jovial field Sep 20, 2024, 5:39 PM

#

We can discuss this later

patent raptor Sep 20, 2024, 5:39 PM

#

They are so close

#

that the quotient of n! and that formula is 1

#

that's how close they are

jovial field Sep 20, 2024, 5:39 PM

#

You are right

patent raptor Sep 20, 2024, 5:39 PM

#

It's like comparing

#

n and n+1

jovial field Sep 20, 2024, 5:39 PM

#

But the difficulty is that believe me

#

Its not that type of the assignment where u need Stirling

patent raptor Sep 20, 2024, 5:40 PM

#

,w limit ( (e^(2n))/(pi4^nn^(2n+1) ))^(1/n) as n -> inf

grizzled pagodaBOT Sep 20, 2024, 5:41 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 5:41 PM

#

root test implies it converges

patent raptor Sep 20, 2024, 5:41 PM

#

jovial field Its not that type of the assignment where u need Stirling

wdym

#

you think there is a way more simple way

jovial field Sep 20, 2024, 5:41 PM

#

But root test was not conclusive for our original series

patent raptor Sep 20, 2024, 5:41 PM

#

we tried like 5 tests

jovial field Sep 20, 2024, 5:41 PM

#

Don't u think ?

#

It may cause some issue

patent raptor Sep 20, 2024, 5:41 PM

#

but this one it is

#

Ok then

jovial field Sep 20, 2024, 5:42 PM

#

See

patent raptor Sep 20, 2024, 5:42 PM

#

Use limit comparison test

jovial field Sep 20, 2024, 5:42 PM

#

See

#

We can discuss that later

#

Let's discuss the rest

#

So that I can complete them

#

I leave a page for 15.

#

Ok?

patent raptor Sep 20, 2024, 5:42 PM

#

,w limit [((2n)!)^2/(2^nn!)^4] / [ (e^(2n))/(pi4^nn^(2n+1)) ] as n -> inf

grizzled pagodaBOT Sep 20, 2024, 5:42 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 5:43 PM

#

wtf

jovial field Sep 20, 2024, 5:43 PM

#

Wait

#

Idea

#

,w Lim n tends to infinity (2n!)^2/[2^n*n!]^4

grizzled pagodaBOT Sep 20, 2024, 5:44 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 5:44 PM

#

Shi

patent raptor Sep 20, 2024, 5:44 PM

#

yea

jovial field Sep 20, 2024, 5:44 PM

#

Can't use that nature

patent raptor Sep 20, 2024, 5:44 PM

#

yea

patent raptor Sep 20, 2024, 5:44 PM

#

grizzled pagoda

damn this hurts

#

,w limit (n! - sqrt(2pi*n)(n/e)^n) as n -> inf

grizzled pagodaBOT Sep 20, 2024, 5:45 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 5:45 PM

#

☠️

#

See

#

Bro

#

Let's discuss this later

#

We have 5 more questions

#

16-17-18-19-20

#

sully

#

This question is not meant for this assignment but is given

#

Let's do the rest

patent raptor Sep 20, 2024, 5:46 PM

#

so weird I expected 0

jovial field Sep 20, 2024, 5:46 PM

#

Wait

#

Let's go forward

#

patent raptor Sep 20, 2024, 5:47 PM

#

ok bro

jovial field Sep 20, 2024, 5:47 PM

#

patent raptor Sep 20, 2024, 5:47 PM

#

16

jovial field Sep 20, 2024, 5:47 PM

#

Nth term is x^n(logn)^q

#

Root test?

patent raptor Sep 20, 2024, 5:48 PM

#

,, \sum_{n=2}^\infty x^{n}\log^p n

grizzled pagodaBOT Sep 20, 2024, 5:48 PM

#

bacc

jovial field Sep 20, 2024, 5:48 PM

#

Yes

#

,w lim n tends to infinity (log n)^(1/n)

grizzled pagodaBOT Sep 20, 2024, 5:49 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 5:49 PM

#

Yes

#

But shit

patent raptor Sep 20, 2024, 5:49 PM

#

root test seems inconclusive

jovial field Sep 20, 2024, 5:49 PM

#

Root test failed

patent raptor Sep 20, 2024, 5:49 PM

#

for all p

jovial field Sep 20, 2024, 5:49 PM

#

Oof

patent raptor Sep 20, 2024, 5:49 PM

#

My guess is that |x| < 1

jovial field Sep 20, 2024, 5:50 PM

#

D'alembert?

patent raptor Sep 20, 2024, 5:50 PM

#

grizzled pagoda **bacc**

looking at this

jovial field Sep 20, 2024, 5:50 PM

#

Wait let's see

patent raptor Sep 20, 2024, 5:50 PM

#

oh wait

#

it's a power series

#

right

patent raptor Sep 20, 2024, 5:50 PM

#

grizzled pagoda

Then indeed

jovial field Sep 20, 2024, 5:51 PM

#

,w lim n tends to infinity [log(n)/log(n+1)]^q

patent raptor Sep 20, 2024, 5:51 PM

#

|x| < 1

grizzled pagodaBOT Sep 20, 2024, 5:51 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 5:51 PM

#

Its 1

patent raptor Sep 20, 2024, 5:51 PM

#

ye

jovial field Sep 20, 2024, 5:51 PM

#

Lol

patent raptor Sep 20, 2024, 5:51 PM

#

when it comes to power series

jovial field Sep 20, 2024, 5:51 PM

#

Its infinity by infinity form

#

But shit

patent raptor Sep 20, 2024, 5:51 PM

#

you can use these test

jovial field Sep 20, 2024, 5:51 PM

#

D'Alembert

#

Is gone

#

No no

#

X will also come

#

Yes

#

Yes

patent raptor Sep 20, 2024, 5:51 PM

#

so for |x| < 1 it converegs

#

done

#

17

jovial field Sep 20, 2024, 5:51 PM

#

So ok for this question I am writing d alembert

#

Wait 1 sec

#

Lemme note the idea

patent raptor Sep 20, 2024, 5:52 PM

#

i think root test is cleaner

jovial field Sep 20, 2024, 5:52 PM

#

Root test

#

Aah

patent raptor Sep 20, 2024, 5:52 PM

#

you can use squeze theorem with n^p

jovial field Sep 20, 2024, 5:52 PM

#

Yes

#

Okok

#

Fr

#

Fr

#

Fr

#

But for x=1 it will fail

#

Then what should I do

patent raptor Sep 20, 2024, 5:52 PM

#

nothing

jovial field Sep 20, 2024, 5:52 PM

#

Which test?

#

To get the nature at x=1

patent raptor Sep 20, 2024, 5:52 PM

#

it fails naturally

#

cause then it is not even a null sequence

#

log(n)^p -> inf as n->inf

#

done

#

unless wait

#

p < 0

jovial field Sep 20, 2024, 5:53 PM

#

No no

#

We don't care

patent raptor Sep 20, 2024, 5:53 PM

#

then you get 1/log(n)

jovial field Sep 20, 2024, 5:53 PM

#

We care bout z

#

x*

patent raptor Sep 20, 2024, 5:54 PM

#

yea log is too slow

#

to make any difference

#

fr

jovial field Sep 20, 2024, 5:54 PM

#

If x=1 series will be [log(n)]^p

patent raptor Sep 20, 2024, 5:54 PM

#

ok

#

so

jovial field Sep 20, 2024, 5:54 PM

#

Now I will use Un

#

Positive term

patent raptor Sep 20, 2024, 5:54 PM

#

,w sum 1/(log(n))^1000

jovial field Sep 20, 2024, 5:54 PM

#

Lim n tend un

grizzled pagodaBOT Sep 20, 2024, 5:54 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 5:54 PM

#

It will diverge

#

Fair

patent raptor Sep 20, 2024, 5:54 PM

#

yea i am dumb

jovial field Sep 20, 2024, 5:55 PM

#

No no

patent raptor Sep 20, 2024, 5:55 PM

#

1/n

#

anyway

jovial field Sep 20, 2024, 5:55 PM

#

So final

patent raptor Sep 20, 2024, 5:55 PM

#

let's move on

jovial field Sep 20, 2024, 5:55 PM

#

Root test for basic

patent raptor Sep 20, 2024, 5:55 PM

#

we got 2 minutes

jovial field Sep 20, 2024, 5:55 PM

#

And Positive term sequence test when x=1

patent raptor Sep 20, 2024, 5:55 PM

#

ye

jovial field Sep 20, 2024, 5:55 PM

#

Okok

#

patent raptor Sep 20, 2024, 5:55 PM

#

can you post it again 17

jovial field Sep 20, 2024, 5:55 PM

#

#

Sure

patent raptor Sep 20, 2024, 5:55 PM

#

thanks g

jovial field Sep 20, 2024, 5:55 PM

#

No probs

patent raptor Sep 20, 2024, 5:56 PM

#

ok 17 is kinda related to 15

#

but it's reciprocal

#

well not quiet

#

it's

#

,, \sum_{n=1}^\infty \frac{(n!)^2}{\left ( \frac{(2(n+1))!}{2^{n+1}(n+1)!} \right )^2}

jovial field Sep 20, 2024, 5:57 PM

#

Oof

patent raptor Sep 20, 2024, 5:57 PM

#

we need to do actually

grizzled pagodaBOT Sep 20, 2024, 5:57 PM

#

bacc

patent raptor Sep 20, 2024, 5:58 PM

#

it starts with 3

jovial field Sep 20, 2024, 5:58 PM

#

Yes

patent raptor Sep 20, 2024, 5:58 PM

#

the odd factor

#

,calc 4!/(2^2(2!))

grizzled pagodaBOT Sep 20, 2024, 5:58 PM

#

Result:

patent raptor Sep 20, 2024, 5:58 PM

#

ye

jovial field Sep 20, 2024, 5:58 PM

#

Yes

patent raptor Sep 20, 2024, 5:58 PM

#

move on?

#

it' kinda the same bs

#

as 15

jovial field Sep 20, 2024, 5:58 PM

#

But

#

See

#

I feel in this root test alembert or rabbe might work

#

Like we might see 1/4 typa shi

patent raptor Sep 20, 2024, 5:59 PM

#

I recommend root test if we have something with power to n

jovial field Sep 20, 2024, 5:59 PM

#

Yes

#

See if I try D'Alembert

patent raptor Sep 20, 2024, 5:59 PM

#

factorials are very good for ratio test

jovial field Sep 20, 2024, 5:59 PM

#

Yes okok

#

Let's check

#

Can you check ?

#

I don't know latex

patent raptor Sep 20, 2024, 6:00 PM

#

ye

#

ratio test works

jovial field Sep 20, 2024, 6:00 PM

#

Yes

#

Done

#

What is the nth term in this

patent raptor Sep 20, 2024, 6:01 PM

#

#

18

jovial field Sep 20, 2024, 6:01 PM

#

Okok

patent raptor Sep 20, 2024, 6:01 PM

#

ok that's easy

jovial field Sep 20, 2024, 6:02 PM

#

patent raptor Sep 20, 2024, 6:02 PM

#

,, \sum_{n=0}^\infty \frac{x^{2n}}{(n+2)\sqrt{n+1}}

jovial field Sep 20, 2024, 6:02 PM

#

x^(2n)

patent raptor Sep 20, 2024, 6:02 PM

#

bet

jovial field Sep 20, 2024, 6:02 PM

#

Yes

grizzled pagodaBOT Sep 20, 2024, 6:03 PM

#

bacc

patent raptor Sep 20, 2024, 6:03 PM

#

probably ratio

jovial field Sep 20, 2024, 6:03 PM

#

Fair

#

I feel

#

Too

patent raptor Sep 20, 2024, 6:03 PM

#

ye

jovial field Sep 20, 2024, 6:03 PM

#

But I feel

patent raptor Sep 20, 2024, 6:03 PM

#

then factor the highest n

jovial field Sep 20, 2024, 6:03 PM

#

Limit form of comparison test

#

If you take common

#

You would get n^3/2

patent raptor Sep 20, 2024, 6:03 PM

#

Wait

#

looking at it

jovial field Sep 20, 2024, 6:03 PM

#

Which would converge

patent raptor Sep 20, 2024, 6:03 PM

#

fr

#

1

jovial field Sep 20, 2024, 6:03 PM

#

By p series

#

So fair

patent raptor Sep 20, 2024, 6:03 PM

#

i forgot it's power series

#

so dont care

jovial field Sep 20, 2024, 6:04 PM

#

Limit form

patent raptor Sep 20, 2024, 6:04 PM

#

|x| < 1

#

easy

jovial field Sep 20, 2024, 6:04 PM

#

Limit form

#

Vn=n^3/2

#

Done

#

patent raptor Sep 20, 2024, 6:04 PM

#

,w limit (1/[(n+2)(n+1)^(1/2)])/(1/[(n+2+1)(n+1+1)^(1/2)]) as n to infinity

jovial field Sep 20, 2024, 6:04 PM

#

grizzled pagodaBOT Sep 20, 2024, 6:04 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 6:04 PM

#

yea

#

x^2 < 1

#

so -1 < x < 1

jovial field Sep 20, 2024, 6:04 PM

#

Yea

patent raptor Sep 20, 2024, 6:04 PM

#

ez

jovial field Sep 20, 2024, 6:05 PM

#

Yea

patent raptor Sep 20, 2024, 6:05 PM

#

ok 19 ez

jovial field Sep 20, 2024, 6:05 PM

#

But wait

#

In 18

patent raptor Sep 20, 2024, 6:05 PM

#

bet

jovial field Sep 20, 2024, 6:05 PM

#

If I apply limit form of comparison test

#

x^n will be in numerator

patent raptor Sep 20, 2024, 6:05 PM

#

yea

jovial field Sep 20, 2024, 6:05 PM

#

I can't apply it

#

So what ?

#

Ratio?

patent raptor Sep 20, 2024, 6:05 PM

#

dont

#

yea

#

ratio

#

goes to 1

#

you can see you will have

#

n^(3/2) .... / n^(3/2)

#

so x^2 < 1

jovial field Sep 20, 2024, 6:06 PM

#

so I will have simply x

#

Yes yes

#

Fair

patent raptor Sep 20, 2024, 6:06 PM

#

fr

jovial field Sep 20, 2024, 6:06 PM

#

patent raptor Sep 20, 2024, 6:07 PM

#

fr

#

they dont sort from easy to difficult

#

they mix it

#

,, \sum_{n=1}^\infty \frac{n}{(2n-1)(2n)}

jovial field Sep 20, 2024, 6:08 PM

#

No

#

See

#

2/(3.4)

grizzled pagodaBOT Sep 20, 2024, 6:08 PM

#

bacc

jovial field Sep 20, 2024, 6:08 PM

#

Fair

patent raptor Sep 20, 2024, 6:08 PM

#

bet

#

i tested you

#

diverges

jovial field Sep 20, 2024, 6:09 PM

#

Ya fair now

patent raptor Sep 20, 2024, 6:09 PM

#

1/n

jovial field Sep 20, 2024, 6:09 PM

#

Ez

patent raptor Sep 20, 2024, 6:09 PM

#

easy

#

bro you cooking

#

fr

jovial field Sep 20, 2024, 6:09 PM

#

Limit form

patent raptor Sep 20, 2024, 6:09 PM

#

🍽️

patent raptor Sep 20, 2024, 6:09 PM

#

jovial field Limit form

comparison test this

jovial field Sep 20, 2024, 6:09 PM

#

Yes

#

It will come 1/n

patent raptor Sep 20, 2024, 6:09 PM

#

ye

jovial field Sep 20, 2024, 6:09 PM

#

I take Vn as 1/n

#

Diverge

#

Wait

#

We have the last one now

patent raptor Sep 20, 2024, 6:10 PM

#

ye

#

which last one

jovial field Sep 20, 2024, 6:11 PM

#

This might be a good one

patent raptor Sep 20, 2024, 6:11 PM

#

bet

jovial field Sep 20, 2024, 6:11 PM

#

#

(logn)^2/(n)^3/2

patent raptor Sep 20, 2024, 6:13 PM

#

bet

jovial field Sep 20, 2024, 6:13 PM

#

Maybe

#

Limit form of?

patent raptor Sep 20, 2024, 6:13 PM

#

ratio test

jovial field Sep 20, 2024, 6:14 PM

#

no cap n^3/2 kinda attracting me

patent raptor Sep 20, 2024, 6:14 PM

#

fr you got weird attrations

jovial field Sep 20, 2024, 6:14 PM

#

Lol

patent raptor Sep 20, 2024, 6:14 PM

#

jk

#

bro

jovial field Sep 20, 2024, 6:14 PM

#

No prob

#

Bro

#

You the 🐐

#

If the 🐐 says one must listen

#

Ratio test

#

Ok

#

I believe on you

#

Fair we did all

#

Now we just need to crack the 15.

#

And we done

#

#

And bro pls also repeat the nth term for 17.

#

Then we done

#

Imma complete it

patent raptor Sep 20, 2024, 6:16 PM

#

somewhere above

patent raptor Sep 20, 2024, 6:16 PM

#

grizzled pagoda **bacc**

.

jovial field Sep 20, 2024, 6:17 PM

#

Fair

#

Ok we got

#

Now only we need to crack 15.

#

And we good to go

#

Then imma fair it out fast

patent raptor Sep 20, 2024, 6:18 PM

#

bet

jovial field Sep 20, 2024, 6:18 PM

#

Yes bet fr

#

16-17-18-19-20 we discussed

patent raptor Sep 20, 2024, 6:18 PM

#

,w limit (n! - sqrt(2pi*n)(n/e)^n+1) as n -> inf

grizzled pagodaBOT Sep 20, 2024, 6:18 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 6:18 PM

#

Just need the 15

#

💀☠️

patent raptor Sep 20, 2024, 6:19 PM

#

grizzled pagoda **bacc**

we were at this

jovial field Sep 20, 2024, 6:19 PM

#

Right

#

But applying approximation

#

And changing the nth term

#

Makes the root test conclusive now

#

This is the problem

#

I feel this will be done by direct comparison only

#

Or limit form

patent raptor Sep 20, 2024, 6:20 PM

#

yea i see

#

you are good

#

let me cook

jovial field Sep 20, 2024, 6:20 PM

#

Fr

#

The cooker 🔥

deft frost Sep 20, 2024, 6:20 PM

#

Sup

jovial field Sep 20, 2024, 6:20 PM

#

Sup

deft frost Sep 20, 2024, 6:20 PM

#

Long time no see

#

Oppailol

jovial field Sep 20, 2024, 6:20 PM

#

Sup

#

Who

deft frost Sep 20, 2024, 6:21 PM

#

You've gotten the active role

jovial field Sep 20, 2024, 6:21 PM

#

Who?

deft frost Sep 20, 2024, 6:21 PM

#

You've grown

jovial field Sep 20, 2024, 6:21 PM

#

Who?

deft frost Sep 20, 2024, 6:21 PM

#

jovial field Who?

Check dm

jovial field Sep 20, 2024, 6:21 PM

#

Yo

#

Wassup

deft frost Sep 20, 2024, 6:22 PM

#

Js fine

#

Came here to check on ya

jovial field Sep 20, 2024, 6:22 PM

#

The goat helping me

deft frost Sep 20, 2024, 6:22 PM

#

Cuz you and kheeri carried me through hs

#

Last year

jovial field Sep 20, 2024, 6:22 PM

#

Fr

#

Ik

#

The 🐐 helping me through an assignment

deft frost Sep 20, 2024, 6:23 PM

#

Aight gl

jovial field Sep 20, 2024, 6:23 PM

#

gl

patent raptor Sep 20, 2024, 6:24 PM

#

Damn

#

maybe you are right using integral test

#

shit

jovial field Sep 20, 2024, 6:24 PM

#

Nooooooo

#

I am ded

#

What they want us to use gamma?

patent raptor Sep 20, 2024, 6:25 PM

#

fr

jovial field Sep 20, 2024, 6:25 PM

#

No

#

💀☠️

#

Na bruh

#

I can't write that 2 page typa shi in the assignment

patent raptor Sep 20, 2024, 6:25 PM

#

,, \sum_{n=1}^{\infty} \frac{[(2n)!]^2}{[2^nn!]^4}

patent raptor Sep 20, 2024, 6:25 PM

#

patent raptor ,, \sum_{n=1}^{\infty} \frac{[(2n)!]^2}{[2^nn!]^4} \approx \sum_{n=1}^{\infty} \...

.

jovial field Sep 20, 2024, 6:25 PM

#

It did not

#

Do

grizzled pagodaBOT Sep 20, 2024, 6:25 PM

#

bacc

jovial field Sep 20, 2024, 6:25 PM

#

Oof

patent raptor Sep 20, 2024, 6:25 PM

#

Let's see if we can

#

simplify more

jovial field Sep 20, 2024, 6:25 PM

#

Okok

#

Log test does not work else we were good to gi

#

Go

patent raptor Sep 20, 2024, 6:27 PM

#

,, \sum_{n=1}^{\infty} \left ( \frac{(2n)!}{2^{2n} n! \cdot n!} \right )^2 = \sum_{n=1}^{\infty} \left ( \frac{1 \cdot 2 \cdot ... \cdot (2n-1) \cdot (2n)}{2^{2n} n! \cdot 1 \cdot 2 \cdot ... \cdot (n-1) \cdot n} \right )^2

grizzled pagodaBOT Sep 20, 2024, 6:28 PM

#

bacc

jovial field Sep 20, 2024, 6:28 PM

#

Yea

patent raptor Sep 20, 2024, 6:29 PM

#

Actually

#

n! < n^n

jovial field Sep 20, 2024, 6:29 PM

#

Yes

#

Yes

#

So

#

Si

#

So*

patent raptor Sep 20, 2024, 6:29 PM

#

(2n)! < (2n)^(2n)

#

let's try that

#

and root test this

jovial field Sep 20, 2024, 6:29 PM

#

Fr

#

Fr

patent raptor Sep 20, 2024, 6:30 PM

#

,, \sum_{n=1}^{\infty} \left ( \frac{(2n)!}{2^{2n} n! \cdot n!} \right )^2 \leq \sum_{n=1}^{\infty} \left ( \frac{(2n)^{2n}}{2^{2n} \cdot (n!)^2} \right )^2

grizzled pagodaBOT Sep 20, 2024, 6:30 PM

#

bacc

patent raptor Sep 20, 2024, 6:31 PM

#

,w limit ([(2n)^(2n)]/(2^(2n)*(n!)^2))^(2/n) n -> inf

grizzled pagodaBOT Sep 20, 2024, 6:31 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 6:32 PM

#

So this diverges

patent raptor Sep 20, 2024, 6:32 PM

#

yea

jovial field Sep 20, 2024, 6:32 PM

#

Positive term series

patent raptor Sep 20, 2024, 6:32 PM

#

no

jovial field Sep 20, 2024, 6:32 PM

#

Diverges

#

Does not?

patent raptor Sep 20, 2024, 6:32 PM

#

this just means

jovial field Sep 20, 2024, 6:32 PM

#

Oh shi I did not see 1/n

patent raptor Sep 20, 2024, 6:32 PM

#

we found a too big upper bound

jovial field Sep 20, 2024, 6:32 PM

#

Got it

patent raptor Sep 20, 2024, 6:32 PM

#

the comparison test would be valid if a greater series converges then ofc the smaller has too

jovial field Sep 20, 2024, 6:33 PM

#

Yes

patent raptor Sep 20, 2024, 6:34 PM

#

damn

#

,w limit ( (2n)!/( 2^(2n)*(n^(2n))) )^(2/n) n -> inf

grizzled pagodaBOT Sep 20, 2024, 6:35 PM

#

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

patent raptor Sep 20, 2024, 6:35 PM

#

bro

jovial field Sep 20, 2024, 6:35 PM

#

Yes

patent raptor Sep 20, 2024, 6:35 PM

#

ok comparison test won't work

#

this goes to 1/e^4

jovial field Sep 20, 2024, 6:35 PM

#

Ik

#

I saw it now

#

patent raptor Sep 20, 2024, 6:36 PM

#

one last test

jovial field Sep 20, 2024, 6:36 PM

#

Ok

patent raptor Sep 20, 2024, 6:36 PM

#

,w limit ([(2n)^(2n)]/(2^(2n)*(n^n)^2))^(2/n) n -> inf

grizzled pagodaBOT Sep 20, 2024, 6:36 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 6:36 PM

#

haha

#

bet

jovial field Sep 20, 2024, 6:36 PM

#

Lol

#

Fails fr

#

What if we break

#

And identify Vn

#

Like we take time

#

Then

patent raptor Sep 20, 2024, 6:37 PM

#

What is Vn

jovial field Sep 20, 2024, 6:38 PM

#

Auxiliary series

patent raptor Sep 20, 2024, 6:38 PM

#

p series test?

jovial field Sep 20, 2024, 6:38 PM

#

See

#

I want to take some part out of this n term

#

For which we have a known series

patent raptor Sep 20, 2024, 6:39 PM

#

,, \sum_{n=1}^{\infty} \left ( \frac{(2n)!}{2^{2n} n! \cdot n!} \right )^2 = \sum_{n=1}^{\infty} \left ( \frac{1 \cdot 2 \cdot ... \cdot (2n-1) \cdot (2n)}{2^{2n} \cdot 1^2 \cdot 2^2 \cdot ... \cdot (n-1)^2 \cdot n^2} \right )^2

jovial field Sep 20, 2024, 6:39 PM

#

And we compare this with that

grizzled pagodaBOT Sep 20, 2024, 6:39 PM

#

bacc

patent raptor Sep 20, 2024, 6:39 PM

#

,w (2n)!/(n!)^2

#

bet let's use the gammoto function

jovial field Sep 20, 2024, 6:40 PM

#

What

patent raptor Sep 20, 2024, 6:40 PM

#

gamma function

jovial field Sep 20, 2024, 6:40 PM

#

Oof

#

Okok

#

Fair

#

We have nth term

patent raptor Sep 20, 2024, 6:41 PM

#

,w (2^(2 n) Γ(1/2 + n))/(sqrt(π) Γ(1 + n))=(2n)!/(n!)^2

grizzled pagodaBOT Sep 20, 2024, 6:41 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 6:41 PM

#

2^2n cancels

#

gamma/gamm basically

jovial field Sep 20, 2024, 6:41 PM

#

Yess

patent raptor Sep 20, 2024, 6:42 PM

#

,, \sum_{n=1}^{\infty} \left ( \frac{(2n)!}{2^{2n} n! \cdot n!} \right )^2 = \frac{\sqrt{\pi}}{\pi} \sum_{n=1}^{\infty} \left ( \frac{\Gamma (\frac{1}{2}+n )}{\Gamma (1+n )} \right )^2

#

bet

jovial field Sep 20, 2024, 6:42 PM

#

bet

#

Search on

#

Yt

#

Some similar thing has been evaluated by maths50t

#

Maths505*

patent raptor Sep 20, 2024, 6:43 PM

#

nah

#

gamma is increasing

jovial field Sep 20, 2024, 6:44 PM

#

Nah

#

Its different

#

Wait

#

Idea

patent raptor Sep 20, 2024, 6:44 PM

#

nah i have it

jovial field Sep 20, 2024, 6:44 PM

#

See multiply and divide by gamma(1/2) in num and denom

#

It becomes beta(1/2,1/2+k)/gamma(1/2)

patent raptor Sep 20, 2024, 6:45 PM

#

ratio test of that goes to 0

#

so it converges

#

done

jovial field Sep 20, 2024, 6:45 PM

#

Lol

#

Is this shi

#

Even acceptwd

patent raptor Sep 20, 2024, 6:45 PM

#

wait

#

wtf

#

ratio test works but it's not a null sequence

#

,w limit of Gamma(2n)/Gamma(1 + n) as n -> inf

grizzled pagodaBOT Sep 20, 2024, 6:48 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 6:48 PM

#

bruh

jovial field Sep 20, 2024, 6:48 PM

#

Aah

grizzled pagodaBOT Sep 20, 2024, 6:48 PM

#

bacc

jovial field Sep 20, 2024, 6:48 PM

#

Search

#

On net

patent raptor Sep 20, 2024, 6:48 PM

#

We need to use properties of gamma

#

I know them

jovial field Sep 20, 2024, 6:48 PM

#

Bro the term inside the bracket is literally

#

Beta(1/2,1/2+k)/gamma(1/2)

#

Gamma(1/2) take it out now you just have beta

patent raptor Sep 20, 2024, 6:51 PM

#

,, \frac{\sqrt{\pi}}{\pi} \sum_{n=1}^{\infty} \left ( \frac{\Gamma (\frac{1}{2}+n )}{\Gamma (1+n )} \right )^2 =
\frac{\sqrt{\pi}}{\pi} \sum_{n=1}^{\infty} \left ( \frac{\Gamma (\frac{1}{2} + n )}{n\Gamma (n)} \right )^2

grizzled pagodaBOT Sep 20, 2024, 6:51 PM

#

bacc

jovial field Sep 20, 2024, 6:51 PM

#

Bro

patent raptor Sep 20, 2024, 6:52 PM

#

wait hold on

jovial field Sep 20, 2024, 6:53 PM

#

Sure

patent raptor Sep 20, 2024, 6:53 PM

#

jovial field Beta(1/2,1/2+k)/gamma(1/2)

how

jovial field Sep 20, 2024, 6:54 PM

#

Beta(1/2,1/2+k)=gamma(1/2)gamma(1/2+k) whole by gamma(1/2+1/2+k) which is gamma(1+k)

#

Now see you have this only

#

In the bracket just multiply divide by gamma(1/2)

patent raptor Sep 20, 2024, 6:55 PM

#

ok

#

,, \frac{\sqrt{\pi}}{\pi} \sum_{n=1}^{\infty} \left ( \frac{\Gamma(\frac{1}{2})\Gamma (n+\frac{1}{2})}{\Gamma(\frac{1}{2})\Gamma (n+1)} \right )^2

grizzled pagodaBOT Sep 20, 2024, 6:55 PM

#

bacc

jovial field Sep 20, 2024, 6:55 PM

#

Yes

patent raptor Sep 20, 2024, 6:56 PM

#

ahhhh

#

you smart af

jovial field Sep 20, 2024, 6:56 PM

#

Yes

#

I have said this 2 times

#

Now see if this would help

#

💀☠️

patent raptor Sep 20, 2024, 6:56 PM

#

sorry lord oppailol

jovial field Sep 20, 2024, 6:56 PM

#

No

#

Lord Adonis you the 🐐

#

And we know gamma(1/2)=√π

grizzled pagodaBOT Sep 20, 2024, 6:58 PM

#

bacc

patent raptor Sep 20, 2024, 6:58 PM

#

,, \frac{\sqrt{\pi}}{\pi^2} \sum_{n=1}^{\infty} \beta \left (\frac{1}{2},n \right )^2

grizzled pagodaBOT Sep 20, 2024, 6:58 PM

#

bacc

patent raptor Sep 20, 2024, 6:59 PM

#

no surprise

#

ratio test inconclusive

jovial field Sep 20, 2024, 6:59 PM

#

Now?

patent raptor Sep 20, 2024, 6:59 PM

#

but it's still a null sequence

jovial field Sep 20, 2024, 6:59 PM

#

💀☠️

patent raptor Sep 20, 2024, 6:59 PM

#

jovial field Now?

no it's good

#

we expected this

#

cause we did equal steps

jovial field Sep 20, 2024, 6:59 PM

#

Yes

#

Now can we do something

patent raptor Sep 20, 2024, 7:00 PM

#

If we do integral test

#

we have a double integral

#

😂

jovial field Sep 20, 2024, 7:00 PM

#

Lol

patent raptor Sep 20, 2024, 7:00 PM

#

and see if that exists

jovial field Sep 20, 2024, 7:00 PM

#

Bro

patent raptor Sep 20, 2024, 7:01 PM

#

nah

jovial field Sep 20, 2024, 7:01 PM

#

How imma write some 3 page typa shi for a question

#

Aah 💀

patent raptor Sep 20, 2024, 7:01 PM

#

we are close

jovial field Sep 20, 2024, 7:01 PM

#

Ok

#

But bro

#

A question

patent raptor Sep 20, 2024, 7:02 PM

#

ok

jovial field Sep 20, 2024, 7:02 PM

#

Think logically

patent raptor Sep 20, 2024, 7:02 PM

#

💀

jovial field Sep 20, 2024, 7:02 PM

#

In a assignment of convergence

#

Why would we have double integral including beta

#

Aah 💀☠️

patent raptor Sep 20, 2024, 7:02 PM

#

bro

#

this was your idea

#

you think

#

logically

jovial field Sep 20, 2024, 7:02 PM

#

Bro there should be some way

#

Else

patent raptor Sep 20, 2024, 7:02 PM

#

💀☠️

jovial field Sep 20, 2024, 7:02 PM

#

patent raptor logically

It was the only way 💀☠️

#

The problem is because of this I can't do other questions

#

This hindering me

patent raptor Sep 20, 2024, 7:03 PM

#

,w integral Beta(1/2,x)^2 from x=1 to x=inf

jovial field Sep 20, 2024, 7:03 PM

#

Lol

#

Type Beta

#

,w int Beta(1/2,x)^2 from 1 to inf

grizzled pagodaBOT Sep 20, 2024, 7:04 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 7:04 PM

#

so it diverges

jovial field Sep 20, 2024, 7:04 PM

#

Does not converge

patent raptor Sep 20, 2024, 7:04 PM

#

by integral test

jovial field Sep 20, 2024, 7:04 PM

#

Hence

#

Series is divergent

#

💀☠️

patent raptor Sep 20, 2024, 7:04 PM

#

now on paper

#

good luck

#

bro

jovial field Sep 20, 2024, 7:04 PM

#

No

#

No bro

patent raptor Sep 20, 2024, 7:04 PM

#

😂

jovial field Sep 20, 2024, 7:04 PM

#

I am cooked

patent raptor Sep 20, 2024, 7:04 PM

#

you cooked

jovial field Sep 20, 2024, 7:05 PM

#

How I am supposed to summarise this whole chat on paper

patent raptor Sep 20, 2024, 7:05 PM

#

deriving divergence with beta function

jovial field Sep 20, 2024, 7:05 PM

#

💀☠️

jovial field Sep 20, 2024, 7:05 PM

#

patent raptor deriving divergence with beta function

Fr

patent raptor Sep 20, 2024, 7:05 PM

#

beta function derivation is one step

#

then use integral test

#

done

jovial field Sep 20, 2024, 7:05 PM

#

Always 2 steps ahead

patent raptor Sep 20, 2024, 7:05 PM

#

lmao

jovial field Sep 20, 2024, 7:05 PM

#

Lol

#

Bro pls

patent raptor Sep 20, 2024, 7:05 PM

#

You

#

know

#

you are right

#

B^2 is fucked uo

jovial field Sep 20, 2024, 7:05 PM

#

Help

patent raptor Sep 20, 2024, 7:05 PM

#

you integrate a squared integral

jovial field Sep 20, 2024, 7:05 PM

#

I will be killed by a professor

patent raptor Sep 20, 2024, 7:05 PM

#

jovial field Help

!help

amber waspBOT Sep 20, 2024, 7:06 PM

#

jovial field Help

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

jovial field Sep 20, 2024, 7:06 PM

#

If I do this

#

Lol

patent raptor Sep 20, 2024, 7:06 PM

#

nah

#

you are unique bro

jovial field Sep 20, 2024, 7:06 PM

#

Na bro

#

💀☠️

#

Bro can we do some cheating

#

Bro search on the net

patent raptor Sep 20, 2024, 7:06 PM

#

we already did

jovial field Sep 20, 2024, 7:06 PM

#

Its enough

#

No search the question

#

They will do some wrong shit

#

And get the answer

#

I will copy it lmao

#

Bro but yourself thino

#

Think*

patent raptor Sep 20, 2024, 7:07 PM

#

I have an idea

jovial field Sep 20, 2024, 7:07 PM

#

Yes yay

#

Woohoo

patent raptor Sep 20, 2024, 7:07 PM

#

,, \frac{\sqrt{\pi}}{\pi^2} \sum_{n=1}^{\infty} \beta \left (\frac{1}{2},n \right )^2 > \frac{\sqrt{\pi}}{\pi^2} \sum_{n=1}^{\infty} \frac{1}{n}

jovial field Sep 20, 2024, 7:07 PM

#

Proceed lord

#

No

#

Lmao

patent raptor Sep 20, 2024, 7:07 PM

#

ah no wrong

jovial field Sep 20, 2024, 7:07 PM

#

Bro still using Beta

#

💀☠️

#

Alpha?

patent raptor Sep 20, 2024, 7:08 PM

#

wait

grizzled pagodaBOT Sep 20, 2024, 7:09 PM

#

bacc

patent raptor Sep 20, 2024, 7:09 PM

#

try to prove this

jovial field Sep 20, 2024, 7:09 PM

#

💀

patent raptor Sep 20, 2024, 7:09 PM

#

then done

jovial field Sep 20, 2024, 7:09 PM

#

This is tough than the question itself

patent raptor Sep 20, 2024, 7:09 PM

#

jovial field Sep 20, 2024, 7:09 PM

#

Genuine question

patent raptor Sep 20, 2024, 7:09 PM

#

shoot

jovial field Sep 20, 2024, 7:10 PM

#

patent raptor Sep 20, 2024, 7:10 PM

#

,w Beta(1/2,x) > 1/x

jovial field Sep 20, 2024, 7:10 PM

#

Should I leave 1 page for

#

This question

#

1 full page

patent raptor Sep 20, 2024, 7:10 PM

#

jovial field

☠️

patent raptor Sep 20, 2024, 7:10 PM

#

jovial field Should I leave 1 page for

leave 1 page of tear drops

jovial field Sep 20, 2024, 7:11 PM

#

Assignment sheet bro

jovial field Sep 20, 2024, 7:11 PM

#

patent raptor leave 1 page of tear drops

Lmfao

#

I am cooked tomorrow is the final deadline it will not extend now 💀☠️

#

And I was not able to do this question

#

F it

#

I leave one whole page

#

Then I do 16-17-18-19-20

patent raptor Sep 20, 2024, 7:12 PM

#

,, \beta \left (\frac{1}{2},n \right )^2 = \left (\int_0^1 \frac{1}{t^{1/2}}\cdot (1-t)^{n-1} \dd t \right )^2

grizzled pagodaBOT Sep 20, 2024, 7:12 PM

#

bacc

jovial field Sep 20, 2024, 7:12 PM

#

Lol

patent raptor Sep 20, 2024, 7:12 PM

#

this could work

jovial field Sep 20, 2024, 7:12 PM

#

This could work

#

But to make it work

#

You have to work hard

patent raptor Sep 20, 2024, 7:13 PM

#

,w integral (1-t)^(n-1)/sqrt(t) from 0 to 1

#

just goes back to the gamma function

jovial field Sep 20, 2024, 7:14 PM

#

💀

#

Imagine

#

Such type of a question in an assignment which is considered easy

#

The question is wrong

patent raptor Sep 20, 2024, 7:14 PM

#

you are wrong

jovial field Sep 20, 2024, 7:14 PM

#

Bro

#

We tried every test known

#

💀☠️

patent raptor Sep 20, 2024, 7:16 PM

#

LMAO

#

you are so fucking funny

jovial field Sep 20, 2024, 7:16 PM

#

No prob

#

patent raptor Sep 20, 2024, 7:16 PM

#

😂

jovial field Sep 20, 2024, 7:16 PM

#

I am leaving a page for it

patent raptor Sep 20, 2024, 7:16 PM

#

skewed ass sheet

jovial field Sep 20, 2024, 7:16 PM

#

Tomorrow will be a debate

patent raptor Sep 20, 2024, 7:16 PM

#

is it that bad if you wont complete just 1

jovial field Sep 20, 2024, 7:16 PM

#

Between me and him

patent raptor Sep 20, 2024, 7:16 PM

#

actually 2

#

cause

#

17 too

jovial field Sep 20, 2024, 7:17 PM

#

Na 17 is chill

patent raptor Sep 20, 2024, 7:17 PM

#

jovial field Between me and him

pls tell me

jovial field Sep 20, 2024, 7:17 PM

#

We got nth term

#

And ratio test

patent raptor Sep 20, 2024, 7:17 PM

#

jovial field We got nth term

did we?

jovial field Sep 20, 2024, 7:17 PM

#

Yes

patent raptor Sep 20, 2024, 7:17 PM

#

i thought we skipped

jovial field Sep 20, 2024, 7:17 PM

#

You gave it to me

#

Send me again

patent raptor Sep 20, 2024, 7:17 PM

#

dont remember

jovial field Sep 20, 2024, 7:18 PM

#

💀☠️

#

Don't do this to me

patent raptor Sep 20, 2024, 7:19 PM

#

jovial field Yes

i said it lol?

jovial field Sep 20, 2024, 7:19 PM

#

Yes

#

You typed in latex

#

,w Lim n tends to infinity (log n)^(q/n)

#

No

grizzled pagodaBOT Sep 20, 2024, 7:21 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 7:22 PM

#

diverges?

jovial field Sep 20, 2024, 7:22 PM

#

17

#

Aah

#

Maybe?

#

I don't know

#

I have not done it yet

#

Just did 16

patent raptor Sep 20, 2024, 7:23 PM

#

ah yea think limit was 4

jovial field Sep 20, 2024, 7:23 PM

#

Yes

patent raptor Sep 20, 2024, 7:24 PM

#

patent raptor

k bet

#

yea then only 15

#

good luck

jovial field Sep 20, 2024, 7:24 PM

#

I left a page for it

#

💀💀

patent raptor Sep 20, 2024, 7:25 PM

#

bro lacks pages

#

bro

#

we have to do this

jovial field Sep 20, 2024, 7:25 PM

#

No actually

#

See

patent raptor Sep 20, 2024, 7:25 PM

#

you are my hero

jovial field Sep 20, 2024, 7:25 PM

#

And you are my lord

#

Fr

#

The 🐐

patent raptor Sep 20, 2024, 7:26 PM

#

😂

jovial field Sep 20, 2024, 7:26 PM

#

See every question done till now has taken Max to max 1.5 pages

#

That's why I left 1 page accordingly

#

I can't do in 3 pages

#

A single question

#

@patent raptor

#

Give the nth term

#

Pls

#

patent raptor Sep 20, 2024, 7:28 PM

#

huh

patent raptor Sep 20, 2024, 7:29 PM

#

grizzled pagoda **bacc**

.

jovial field Sep 20, 2024, 7:29 PM

#

Ok

#

It is 2(n+1)

#

Or 2n+1

patent raptor Sep 20, 2024, 7:30 PM

#

2(n+1)

jovial field Sep 20, 2024, 7:30 PM

#

Okok

#

Everywhere?

patent raptor Sep 20, 2024, 7:31 PM

#

yea

jovial field Sep 20, 2024, 7:31 PM

#

In power 2 ?

patent raptor Sep 20, 2024, 7:31 PM

#

we said it starts at 3 so n+1 instead n

jovial field Sep 20, 2024, 7:31 PM

#

In power of 2 it is also n+1?

patent raptor Sep 20, 2024, 7:31 PM

#

,calc (1/2)^2

grizzled pagodaBOT Sep 20, 2024, 7:31 PM

#

Result:

0.25

jovial field Sep 20, 2024, 7:31 PM

#

Okok

#

Fair

patent raptor Sep 20, 2024, 7:31 PM

#

,calc (13)^2/(24)^2

grizzled pagodaBOT Sep 20, 2024, 7:31 PM

#

Result:

0.140625

patent raptor Sep 20, 2024, 7:32 PM

#

,, \left ( \frac{1}{2} \right )^2 + \left ( \frac{3}{8} \right )^2 + \left ( \frac{5}{16} \right )^2 +\left ( \frac{35}{128} \right )^2 + ...

grizzled pagodaBOT Sep 20, 2024, 7:35 PM

#

bacc

patent raptor Sep 20, 2024, 7:35 PM

#

I had hopes to detect a pattern

jovial field Sep 20, 2024, 7:36 PM

#

So what happened?

#

Were hopes fulfilled?

patent raptor Sep 20, 2024, 7:36 PM

#

nothing i analyzed 15

jovial field Sep 20, 2024, 7:36 PM

#

Oof

patent raptor Sep 20, 2024, 7:36 PM

#

grizzled pagoda **bacc**

can you see a pattern

jovial field Sep 20, 2024, 7:36 PM

#

In the denominator

patent raptor Sep 20, 2024, 7:36 PM

#

Do you only do sums?

#

or also products

jovial field Sep 20, 2024, 7:37 PM

#

Anything is fine

patent raptor Sep 20, 2024, 7:38 PM

#

$\textbf{15.}\$
[ \sum_{n=1}^\infty \prod_{k=1}^n \left ( \frac{2k-1}{2k} \right )^2 ]

grizzled pagodaBOT Sep 20, 2024, 7:38 PM

#

bacc

patent raptor Sep 20, 2024, 7:39 PM

#

Now the question is if we can do anything with products

#

,w prod (2k-1)^2/(2k)^2 from k = 1 to n

grizzled pagodaBOT Sep 20, 2024, 7:39 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 7:39 PM

#

Lmao

#

😭

#

nah 15 is weird

#

exactly what we derived

jovial field Sep 20, 2024, 7:40 PM

#

Ik

#

The most probable answer is

#

The question is wrong

#

It was not supposed to be this

patent raptor Sep 20, 2024, 7:41 PM

#

but at least

#

at least we found out it's diveregnt

#

because of the beta function

#

graphically speaking

jovial field Sep 20, 2024, 7:41 PM

#

Yes

#

It was good

patent raptor Sep 20, 2024, 7:42 PM

#

i would honestly do it like this with the beta function and argue that graphically 1/n is less and by comparison test it is divergent

jovial field Sep 20, 2024, 7:42 PM

#

Bro can u check D'Alembert for 17

#

Its too confusing

patent raptor Sep 20, 2024, 7:42 PM

#

supposed to be 4

jovial field Sep 20, 2024, 7:42 PM

#

Too hectic

#

4 or 1/4

patent raptor Sep 20, 2024, 7:43 PM

#

I did a_n/a_n+1

#

so by yours 1/4

jovial field Sep 20, 2024, 7:43 PM

#

Okok

patent raptor Sep 20, 2024, 7:43 PM

#

i assumed you did a_n+1 / a_n

jovial field Sep 20, 2024, 7:44 PM

#

Oof

#

3 more left

#

Then bye

#

Not to you

#

This assignment fr lol

patent raptor Sep 20, 2024, 7:44 PM

#

fr

jovial field Sep 20, 2024, 7:44 PM

#

15 is impossible

patent raptor Sep 20, 2024, 7:44 PM

#

calc 2 rn?

jovial field Sep 20, 2024, 7:44 PM

#

Without higher implementation

patent raptor Sep 20, 2024, 7:44 PM

#

ye

#

but you cannot fail because of 1

#

especially if it seems flawed

jovial field Sep 20, 2024, 7:44 PM

#

Yes

patent raptor Sep 20, 2024, 7:44 PM

#

and you still managed to derive something

#

bro

jovial field Sep 20, 2024, 7:45 PM

#

Marks will be cut from internal

patent raptor Sep 20, 2024, 7:45 PM

#

you will appear a genius

#

if you are the only one

jovial field Sep 20, 2024, 7:45 PM

#

If I submit incomplete

patent raptor Sep 20, 2024, 7:45 PM

#

except if there is some trick

jovial field Sep 20, 2024, 7:45 PM

#

I am the only honoured one

#

Fr

patent raptor Sep 20, 2024, 7:45 PM

#

i searched internet but no success

jovial field Sep 20, 2024, 7:45 PM

#

17 is convergent

#

Right?

patent raptor Sep 20, 2024, 7:47 PM

#

should be

#

yea

#

1/4

jovial field Sep 20, 2024, 7:48 PM

#

Bro

#

In 18 what was the nth term

#

x^(2n)?

patent raptor Sep 20, 2024, 7:48 PM

#

grizzled pagoda **bacc**

.

jovial field Sep 20, 2024, 7:49 PM

#

Aah right

patent raptor Sep 20, 2024, 7:53 PM

#

,w double factorial

grizzled pagodaBOT Sep 20, 2024, 7:53 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 7:54 PM

#

#

double factorial

jovial field Sep 20, 2024, 7:55 PM

#

💀☠️

patent raptor Sep 20, 2024, 7:55 PM

#

makes sense

#

then it def. diverges

jovial field Sep 20, 2024, 7:55 PM

#

Wait let's see

#

I mma write this derivative and results

#

In a rough page

#

Then will show him

#

Tomorrow

#

I am sure question is wrong

patent raptor Sep 20, 2024, 7:57 PM

#

,, \sum_{n=1}^{\infty} \left ( \frac{(2n)!}{(2^n n!)^2} \right )^2 = \sum_{n=1}^{\infty} \left ( \frac{(2n-1)!!}{2^n n!} \right )^2 > \sum_{n=1}^{\infty} \left ( \frac{(2n-1)!}{2^n n!} \right )^2

#

relax

jovial field Sep 20, 2024, 7:57 PM

#

Ok

patent raptor Sep 20, 2024, 7:57 PM

#

,w limit (2n-1)!/(2^nn!) n -> inf

jovial field Sep 20, 2024, 7:58 PM

#

Oof

#

Is left

#

Only

patent raptor Sep 20, 2024, 7:58 PM

#

do this

grizzled pagodaBOT Sep 20, 2024, 7:58 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

#

bacc

patent raptor Sep 20, 2024, 7:58 PM

#

bruh

#

one line

jovial field Sep 20, 2024, 7:58 PM

#

What

#

Not possible

#

💀☠️

patent raptor Sep 20, 2024, 7:59 PM

#

yes

jovial field Sep 20, 2024, 7:59 PM

#

I have left one page for this

patent raptor Sep 20, 2024, 7:59 PM

#

https://mathworld.wolfram.com/DoubleFactorial.html

jovial field Sep 20, 2024, 7:59 PM

#

Lol

patent raptor Sep 20, 2024, 7:59 PM

#

look it up

jovial field Sep 20, 2024, 7:59 PM

#

That's fine

#

But wait

#

This means that the nth term should tend to infinity right?

patent raptor Sep 20, 2024, 7:59 PM

#

jovial field Sep 20, 2024, 7:59 PM

#

Which should be evident itself from when we were doing

patent raptor Sep 20, 2024, 8:00 PM

#

jovial field This means that the nth term should tend to infinity right?

yea

jovial field Sep 20, 2024, 8:00 PM

#

Then why was it showing 0

patent raptor Sep 20, 2024, 8:00 PM

#

,w limit (2n)!/(2^nn!)^2 n -> inf

grizzled pagodaBOT Sep 20, 2024, 8:01 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 8:01 PM

#

Aah

#

Tf

#

How this incosistency

patent raptor Sep 20, 2024, 8:01 PM

#

,w plot y = (2x)!/(2^x x!)^2 and y = (2x-1)!!/(2^x x!) between x = 0 and x = 10

jovial field Sep 20, 2024, 8:01 PM

#

This is guiding us wrong

patent raptor Sep 20, 2024, 8:03 PM

#

bro

grizzled pagodaBOT Sep 20, 2024, 8:03 PM

#

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

patent raptor Sep 20, 2024, 8:03 PM

#

AHHHH

#

stupid bot

jovial field Sep 20, 2024, 8:04 PM

#

Ok

patent raptor Sep 20, 2024, 8:05 PM

#

Ok i think

#

It's wrong

#

because n!! is not the same as (n!)!

jovial field Sep 20, 2024, 8:05 PM

#

Aah

#

The problem

#

Is in the interpretation

patent raptor Sep 20, 2024, 8:06 PM

#

it's wrong anyway I think then

jovial field Sep 20, 2024, 8:06 PM

#

Same

#

Cz it diverges

#

If you can express simply

#

I can do this in 2 lines

#

Cz now we know it diverges

#

So lim n tend to inf nth term should not be zero

patent raptor Sep 20, 2024, 8:07 PM

#

uhm

#

there are also series where it goes to 0 but still diverges

#

1/n

jovial field Sep 20, 2024, 8:08 PM

#

Oh

#

No see

#

As you checked limit came infinity

#

From double factorial

patent raptor Sep 20, 2024, 8:09 PM

#

,w Sum[((2n)!/(2^(2n)(n!)^2))^2,{n,1,100}]

#

inf doesnt work

#

but yea it diverges 100 %

jovial field Sep 20, 2024, 8:09 PM

#

Aah

#

Fr I am so cooked by this question

jovial field Sep 20, 2024, 8:10 PM

#

patent raptor but yea it diverges 100 %

I simply want to write this

#

Fr

grizzled pagodaBOT Sep 20, 2024, 8:10 PM

#

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

patent raptor Sep 20, 2024, 8:11 PM

#

finally

jovial field Sep 20, 2024, 8:11 PM

#

Wooho

#

Now how to write it

#

@patent raptor in 20 ratio test works

patent raptor Sep 20, 2024, 8:14 PM

#

Either we derive something with this

grizzled pagodaBOT Sep 20, 2024, 8:14 PM

#

bacc

patent raptor Sep 20, 2024, 8:16 PM

#

We can only tell that n! is in (2n)!

#

So if we were to cancel things we would have some terms of (2n)! left

grizzled pagodaBOT Sep 20, 2024, 8:18 PM

#

bacc

patent raptor Sep 20, 2024, 8:20 PM

#

\begin{gather*} = \sum_{n=1}^{\infty} \left ( \frac{(n+1) \cdot (n+2) \cdot ... \cdot (n+n-1) \cdot (n+n)}{2^{2n} n!} \right )^2
\end{gather*}

grizzled pagodaBOT Sep 20, 2024, 8:20 PM

#

bacc

jovial field Sep 20, 2024, 8:21 PM

#

@patent raptor

#

Aah

#

Test failed

#

Ratio test failed in

#

20

#

Aah

patent raptor Sep 20, 2024, 8:22 PM

#

grizzled pagoda **bacc**

I wonder if we wouldn't have technically n^n in the numerator

jovial field Sep 20, 2024, 8:23 PM

#

Ratio test failed

#

In 20.

patent raptor Sep 20, 2024, 8:23 PM

#

bro

jovial field Sep 20, 2024, 8:23 PM

#

Other alternative?

patent raptor Sep 20, 2024, 8:23 PM

#

post 20

jovial field Sep 20, 2024, 8:23 PM

#

#

It failed

#

No

#

Wait I did wrong I feel

patent raptor Sep 20, 2024, 8:24 PM

#

,w limit [ (log(n+1)^2)/((n+1)^(3/2)) ]/[ (log(n)^2)/((n)^(3/2)) ] n ->inf

grizzled pagodaBOT Sep 20, 2024, 8:24 PM

#

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

jovial field Sep 20, 2024, 8:25 PM

#

Its still failing

#

What should we do

#

@patent raptor

#

Idea came

#

Bacc

#

Bacc

#

What if we do integral test

#

?

#

Aah

#

Bro played with me

#

Fr

#

But my guy 1/n^3/n^3/2 will converge

patent raptor Sep 20, 2024, 8:29 PM

#

ln(x) > 1/x^(1/3)
x > 1 > e^(1/x^(1/3))

#

easy

jovial field Sep 20, 2024, 8:30 PM

#

So tell what should I do

#

Tell I am tired

#

Now

patent raptor Sep 20, 2024, 8:30 PM

#

comparison test for 20

#

15 idk

jovial field Sep 20, 2024, 8:30 PM

#

With which

patent raptor Sep 20, 2024, 8:30 PM

#

grizzled pagoda **bacc**

trying to figure it out here

#

1/n^3

#

you got eyes right

jovial field Sep 20, 2024, 8:31 PM

#

Aah

patent raptor Sep 20, 2024, 8:31 PM

#

jovial field Sep 20, 2024, 8:31 PM

#

How

#

Bro Un>Vn

patent raptor Sep 20, 2024, 8:31 PM

#

patent raptor ln(x) > 1/x^(1/3) x > 1 > e^(1/x^(1/3))

.

jovial field Sep 20, 2024, 8:31 PM

#

Vn should diverge

#

But it will not

patent raptor Sep 20, 2024, 8:31 PM

#

yea

#

p test

#

p < 1

#

hence diverges

jovial field Sep 20, 2024, 8:32 PM

#

This is what pls tell?

#

1/n^3/n^3/2

#

This confuses me

patent raptor Sep 20, 2024, 8:32 PM

#

comparison test bro

#

i find a smaller divergent series

jovial field Sep 20, 2024, 8:32 PM

#

No I k 5hat

#

That

patent raptor Sep 20, 2024, 8:33 PM

#

i chose 1/n^3

#

cause it cancels

jovial field Sep 20, 2024, 8:33 PM

#

I want to ask how n^1/2

#

Came

patent raptor Sep 20, 2024, 8:33 PM

#

shit

jovial field Sep 20, 2024, 8:33 PM

#

Yes

#

So what should i do

patent raptor Sep 20, 2024, 8:34 PM

#

should still work with 1/3