Series

[grim dome]

How to find the radius of convergence of this series?

[little venture]

ratio test

[grim dome]

But what will be an here?

[little venture]

(z+2)^(n-1)/((n+1)^3*4^n)

technically i think you're supposed to take the absolute value of that

[grim dome]

So here i did

An+1/an and i got this

Z+2 (n+1)^3/(n+2)^3.4

[little venture]

wouldn't it be divided by 4

oh that's in the denominator

yes that's right

[grim dome]

So what's next?

[little venture]

find limit of (n+1)^3/(n+2)^3

[grim dome]

Should we ignore 4?

Here I got limit is one

[little venture]

yes

if lim(a) = 1 then lim(ab) = lim(b)

so you can remove that factor

[grim dome]

What do you mean?
(Z+2)/4 is the answer

??

[little venture]

yes that's the limit

[grim dome]

Fine, so what we will do now?

[little venture]

so ratio test says absolute value of that must be < 1 for it to converge, and maybe 1

in the case it asks for radius of convergence you dont need to worry about the 1 case

[grim dome]

Yes right. So it will be less than 1?

[little venture]

|(Z+2)/4| < 1

[grim dome]

Right so should I solve it for value z?

[little venture]

yes

that will give you an interval and then you find half the length of the interval to get the radius

[grim dome]

-6 to 2 so here length is 8 so radius is 4?

[little venture]

seems right

[grim dome]

Thanks

can you tell me any other question related to it?

[little venture]

yes i can help

[grim dome]

I meant like endpoints

[little venture]

oh ic ic

so if it says "find the interval of convergence" or "find all z where this converges" or something

then you would have to plug the endpoints in

[grim dome]

does it not converge between -6 to 2?

[little venture]

but what about at z = -6 and z = 2

ratio test doesn't say

you need a different test

[grim dome]

I see, so what is that test called?

[little venture]

well it depends

you plug those 2 in and you see

what test should i use

[grim dome]

so here we are checking of endpoints for-6,2

[little venture]

where

[grim dome]

so we need to put -6,2 back into the question?

[little venture]

yea

[grim dome]

is there any idea z=1,-1 but putting it and find the radius?

[little venture]

wdym? for example this one the endpoints work out to 1,-1

[grim dome]

The solution in the book, all questions are done with putting z=1 and finding ratio test

So i am asking why are they putting 1 randomly

[little venture]

well, that's pretty dumb

do ratio test first, then plug in the endpoints

that the ratio test doesn't say anything about

[grim dome]

Do you understand Hindi?@little venture

[little venture]

no

[grim dome]

May I know what books you read for this series?

[little venture]

i didnt read any books i just learned what i said in school

[grim dome]

But this level of series is university?

[little venture]

¯_(ツ)_/¯

[grim dome]

Did you learn it at high school?

[little venture]

yes i am a high school student

[grim dome]

May I know your high school board name?

[little venture]

board?

[grim dome]

Who conduct examination

[little venture]

the college board

[grim dome]

And decided the curriculum

[little venture]

ap

[grim dome]

Yes college board

Can you send me link of it?

So that i can compare their syllabus

[little venture]

https://www.collegeboard.org/

https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-bc-course-at-a-glance_0.pdf

[grim dome]

Damn

They included university level things in it

It will be interesting to see what is their university syllabus then😂

[winter moat]

It's almost same for us too. . Except last two ig

[grim dome]

No bro. We don't have such deep syllabus in 11,12th?

Have you ever read convergence in JEE course?

Radius of convergence?

[little venture]

i think they mostly do high school

also why do you need convergence of series for engineering

[grim dome]

Ahhh, I am not an engineering student 🤦😪

[little venture]

math student?

[grim dome]

B.sc b.ed graduated

[little venture]

bachelor of science and education?

[grim dome]

Yes. Different course

Here they are putting random values of z so that it became 1

[grim dome]

See they took x=3

Without taking random values

[winter moat]

Woh hi toh. Except last two chapters

[little venture]

oh yeah this is right

they plugged in the endpoints

[grim dome]

But it was not asked to put endpoints

[little venture]

when it says interval of convergence you need the endpoints

[grim dome]

I see thanks

[little venture]

whereas for radius of convergence the endpoints change the length of the interval by an infinitesimal amount

so yeah

[feral thicket]

I dont understand the language but u can use Cauchy or D.Alembert law. Maybe the D.Alembert law is better