/sci/ - Science & Math » Thread #13887219

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Anonymous Wed 24 Nov 20:31:55 2021 No.13887219 View Reply Original Report

Quoted By: >>13887220 >>13887361 >>13887528 >>13887556

I've been play with the problem for a while now and have no idea how to solve it, and no idea how to look up a solution
> $/sum_{k=2}^\infty {{/sum_{n=0}^\infty \frac{n^k}{k^n}}^{-1}}$
>The explicit summation is something like:
>1/6 + 8/33 + 81/380 + 512/3535 + 15625/189714
If my Latex is fucked pic related should explain good luck bros

Anonymous

Anonymous Wed 24 Nov 2021 20:32:52 No.13887220 Report

Quoted By: >>13887376

>>13887219
Latex is fucked :/

Anonymous

Anonymous Wed 24 Nov 2021 21:19:47 No.13887361 Report

Quoted By:

>>13887219
I would try finding a power series wich when evaluated at a given x value give the inner series, the I would simply sum the known values of the function and sum, try using $\frac{1}{1-x}$
and its derivatives.

Anonymous

Anonymous Wed 24 Nov 2021 21:23:43 No.13887376 Report

Quoted By: >>13887475

>>13887220
your using slashes instead of backslashes, dummy

Anonymous

Anonymous Wed 24 Nov 2021 21:43:45 No.13887446 Report

Quoted By: >>13887475 >>13887612

WA claims it diverges
https://www.wolframalpha.com/input/?i2d=true&i=Sum%5BDivide%5B1%2CPolylog%5C%2891%29-k%5C%2844%29Divide%5B1%2Ck%5D%5C%2893%29%5D%2C%7Bk%2C2%2C%E2%88%9E%7D%5D

Anonymous

Anonymous Wed 24 Nov 2021 21:53:14 No.13887475 Report

Quoted By: >>13887481 >>13887489 >>13887621

>>13887376
god damn it,
${\sum_{k=2}^\infty({\sum_{n=0}^\infty \frac{n^k}{k^n}}^{-1})$
should be fixed?
>>13887446

I wouldn't think so as after the 9th term each term of the series is less than n^-2 so the series should converge.

Anonymous

Anonymous Wed 24 Nov 2021 21:55:00 No.13887481 Report

Quoted By: >>13887486

>>13887475
$\sum_{k=2}^\infty({\sum_{n=0}^\infty \frac{n^k}{k^n}}^{-1})$
if it's still fucked oh well

Anonymous

Anonymous Wed 24 Nov 2021 21:56:01 No.13887486 Report

Quoted By: >>13887509

>>13887481
-1 should be out side prothesis but there ya go

Anonymous

Anonymous Wed 24 Nov 2021 21:56:14 No.13887489 Report

Quoted By:

>>13887475
You can click on the $T_{E} X$ symbol to check if your post is correct, btw.

Anonymous

Anonymous Wed 24 Nov 2021 22:07:51 No.13887509 Report

Quoted By: >>13887544

>>13887486
as a power? as a subtraction?

not like it matters because neither generates
>1/6 + 8/33 + 81/380 + 512/3535 + 15625/189714

just delete this thread and try again when you have learned to post properly.

Anonymous

Anonymous Wed 24 Nov 2021 22:15:29 No.13887528 Report

Quoted By: >>13887544

>>13887219
You really gotta tell us where the series came from, but either way it doesn't seem to converge
https://www.wolframalpha.com/input/?i2d=true&i=Sum%5B%5C%2840%29Polylog%5C%2891%29-k%5C%2844%29Divide%5B1%2Ck%5D%5C%2893%29-1%5C%2841%29%2C%7Bk%2C2%2C%E2%88%9E%7D%5D

Anonymous

Anonymous Wed 24 Nov 2021 22:20:21 No.13887544 Report

Quoted By: >>13887547 >>13887548 >>13887557

>>13887509
>>13887528
here's the a version that'll maybe make more sense

${\sum_{n=0}^\infty \frac{n^2}{2^n}}^{-1} + {\sum_{n=0}^\infty \frac{n^3}{3^n}}^{-1} + {\sum_{n=0}^\infty \frac{n^4}{4^n}}^{-1} + {\sum_{n=0}^\infty \frac{n^5}{5^n}}^{-1} ...$

Anonymous

Anonymous Wed 24 Nov 2021 22:21:36 No.13887547 Report

Quoted By: >>13887553

>>13887544
It still makes zero sense as the -1 is still gibberish

Anonymous

Anonymous Wed 24 Nov 2021 22:21:37 No.13887548 Report

Quoted By: >>13887553

>>13887544
again the -1 should be out side the series to denote it's the reciprocal of the final result and not each term

Anonymous

Anonymous Wed 24 Nov 2021 22:22:38 No.13887553 Report

Quoted By: >>13887557

>>13887547
see
>>13887548

Anonymous

Anonymous Wed 24 Nov 2021 22:23:49 No.13887556 Report

Quoted By:

>>13887219
0

Anonymous

Anonymous Wed 24 Nov 2021 22:24:53 No.13887557 Report

Quoted By: >>13887571

>>13887544
>>13887553
why are you even trying to use LaTeX when you obviously have no idea what you are doing.

do what everyone else does. write it by hand and take a fucking pic.

Anonymous

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Anonymous Wed 24 Nov 2021 22:31:36 No.13887571 Report

Quoted By: >>13887578 >>13887612

>>13887557
Don’t how much clearly this will make it for you nonetheless here you go buddy

Anonymous

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Anonymous Wed 24 Nov 2021 22:33:50 No.13887578 Report

Quoted By:

>>13887571

Anonymous

Anonymous Wed 24 Nov 2021 22:52:29 No.13887612 Report

Quoted By: >>13887621

>>13887571
That's exactly anon's >>13887446 series, which WA thinks diverges, and I have no reason to suspect wolfram wrong.
The term series have a pretty nasty closed form using Eulerian numbers, which is
$\sum_{n\ge 0} \frac{n^k}{k^n} = Li_{-k}(1/k) = \frac{k}{(k-1)^{k+1}}\sum_{j=0}^{k-1}E(k,j)k^j$ .
Why does OP think the reciprocals of these have convergent sum?

Anonymous

Anonymous Wed 24 Nov 2021 22:56:53 No.13887621 Report

Quoted By:

>>13887612
I don't know if OP has a proof of >>13887475 but if you look at the terms numerically it sure looks like they converge.
For example for k = 100 you get ~ 1.02*10^-91.
https://www.wolframalpha.com/input/?i=1%2FPolylog%5B-100%2C1%2F100%5D

Anonymous

Anonymous Thu 25 Nov 2021 14:43:19 No.13889326 Report

Quoted By:

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