Limits, Sandwich Theorem

[rocky eagle]

How can I solve this? Also need a bit of help with how to solve questions involving sandwich theorem… any video suggestions?

[frail locust]

ik that the answer is x

,w lim n to infty (floor(x)+floor(2x)+...+floor(nx))/(1+2+3+...+n)

[neat timber]

Results provided by WolframAlpha

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

Requested by Just A Smart Brat#7677

[frail locust]

oh interesting

i dont think this is correct though

[rocky eagle]

Yes, but how?

[frail locust]

let x be an integer

which means

the thing is just

$\frac{x+2x+3x+\cdots+nx}{1+2+3+\cdots+n}$

[neat timber]

Lav Lavda

[rocky eagle]

Hmm..

[frail locust]

and if you take x common

[rocky eagle]

Yea,

[frail locust]

then you'll be able to divide the numeratir and denominator

and only x will remain

[rocky eagle]

So, we can assume it to be that?

[frail locust]

see if this thing works for any x, and gives the same answer

that means the result for all intgers should give the same answer

[rocky eagle]

Yea

But still it doesn’t prove all the cases

[frail locust]

i'll try to prove this further if i can and ping you here again

[rocky eagle]

Yeah, sure

[frail locust]

if this the full question?

[rocky eagle]

Yes

[frail locust]

its a very weird one

[rocky eagle]

Ik lol

[frail locust]

i can contradict the answer

its not x either

let x = 0.5

the answer will be 2

which is not x

so

yeah

[frail locust]

use the fact that x-1<[x]<= x

[thorn heath]

Wait what is the greatest integer function

Ceiling function?

[frail locust]

no the floor function

[thorn heath]

Oh ok

What is $\lim_{n\to\infty}\frac{x+2x+3x+\cdots+nx}{1+2+3+\cdots+n}$?

[neat timber]

lightn#3358

[thorn heath]

Oh it’s obviously x

[frail locust]

$$x-1<[x] \leq x$$
$$(x-1)+(2x-1)+...+(nx-1) < [x]+[2x]+[...+[nx] \leq 1+2+..+n$$
$$\frac{xn(n+1)}{2}-n < [x]+[2x]+[3x]+...+[nx] \leq x\frac{n(n+1)}{2}$$
$$\frac{\frac{xn(n+1)}{2}-n}{\frac{n(n+1)}{2}} < \frac{[x]+[2x]+[3x]+...+[nx]}{1+2+..+n} \leq x$$

[neat timber]

pratham

[frail locust]

now just apply the limits

yes its x

[rocky eagle]

Ooh

[rocky eagle]

Thanks

[zealous marlin]

@rocky eagle has given 1 rep to @flat sand

[frail locust]

and i think this wasnt calculated correctly

even this is 0.5

[frail locust]

oh yeah i made a stupid error

[frail locust]

thanks

[zealous marlin]

@jade talon has given 1 rep to @flat sand

[fallow echo]

+close