Talk:List of mathematical series

Untitled

Does anybody know the representation for $\sum _{n=0}^{\infty }z^{n^{2}}$, where |z|<1?

It is ${\frac {1+\vartheta _{3}(0,z)}{2}}.$ See Abramowitz, Stegun, Handbook of Mathematical Functions, (16.27.3)--131.220.161.244 (talk) 10:47, 27 August 2009 (UTC)

Infinite sum

What's that? If a sum is infinite it's probably not even defined. Are we talking about diverging series? 213.112.76.90 (talk) 13:05, 15 May 2009 (UTC)

No, we talking about a sum of infinite object which convergent into a value. see also Infinite sum --77.125.107.105 (talk) 11:57, 30 August 2009 (UTC)

Confusing notation

I am probably going to change the index of summation in all the series on the page to $m$ -- having it as $i$ is probably a good source of unnecessary confusion (conflicts with $i={\sqrt {-1}}$) tryptographer (talk) 06:17, 14 July 2010 (UTC)

Trigonometric functions

These two equations are wrong! If n is a natural number any of those two sums are in general different from zero. If n tends to infinity, the series converge, but not to zero. The sum of cosines would converge if m started on zero...

Please somebody correct me if I'm wrong, otherwise correct the article. —Preceding unsigned comment added by 193.136.104.23 (talk) 11:24, 6 September 2010 (UTC)

Proofs

What about the proofs for some of these closed forms? I there a place I'd be able to find that?

I, too, would value a reference for the closed form expressions. I can hardly name Wikipedia as a reference. — Preceding unsigned comment added by 141.211.139.138 (talk) 00:20, 8 April 2012 (UTC)

Sepating into finite series and infinite series

What do you think about separating this article into finite series and infinite series?--MathFacts (talk) 06:00, 10 November 2010 (UTC)

Finite Sum Power Series

Where did the formulas for series like $\sum _{m=1}^{n}m^{2}x^{m}={\frac {x(1+x-(n+1)^{2}x^{n}+(2n^{2}+2n-1)x^{n+1}-n^{2}x^{n+2})}{(1-x)^{3}}}\,\!$ come from? Is there a solution for $\sum _{m=1}^{n}m^{k}x^{m}\!$? — Preceding unsigned comment added by BenRMorin (talk • contribs) 07:52, 22 March 2011 (UTC)

the solution for $\sum _{m=0}^{n-1}m^{k}x^{m}\!$ is $(xd/dx)^{k}(1-x^{n})/(1-x)$, where $(xd/dx)^{k}$ is the operator (x times derivative(x)) applied k times. --131.220.161.244 (talk) 14:48, 2 May 2012 (UTC)

Geometric Progression of 'i' i.e. S= 1+i+i²+i³+i⁴+... Upto N terms

Ubove Sum S can only have following 4 values for every N being Even Or Odd

1. S=0

2. S=i+1 3. S=i 4. S=1 No other then these sum exists. Sum upto infinity i.e. S= 1+i+i²+i³+...upto ∞ is ( i+1)/2

Above stated sum can be obtained By using following formulae

S=1+i+i²+i³+...upto N terms

S= \frac{ i+1 }{ 2 } (1+(-1) ^ { \frac{ n+2 }{ 2 } } ) if n is Even

n is number of terms in series

S= \frac{ (-1) ^ { \frac{ n-1 }{ 2 { } +(-1) ^ { \frac{ n+1 }{ 2 } } i+i+1 }{ 2 } if n is odd And also it is the average of all the possible sum of GP of i upto N terms. NihalHanfi (talk) 09:42, 17 November 2020 (UTC)

Exponential and logarithmic sum error

Your sum (linked below?) equals ln2-1/2, not ln2.

John W. Patterson (520) 971-4614

https://wikimedia.org/api/rest_v1/media/math/render/svg/940b1790cbab3f08e97e6bbb6559c627404b1806 68.98.225.139 (talk) 22:21, 16 April 2023 (UTC)

Harmonic numbers summation formula is wrong

$\sum _{n=0}^{\infty }{\frac {x^{2}}{n^{2}(n+x)}}=x{\frac {\pi ^{2}}{6}}-H(x)$

is wrong, can't divide by zero $(n=0)$ , it should be

$\sum _{n=1}^{\infty }{\frac {x^{2}}{n^{2}(n+x)}}=x{\frac {\pi ^{2}}{6}}-H(x)$

Could someone verify the correct formula and change it? Potato Imaginator (talk) 03:34, 14 August 2024 (UTC)

pi squared over 8

How about the sum 1/(2k+1)^2 = 1/1^2 + 1/3^2 + 1/5^2 + ... being pi squared over eight?

2601:247:4582:2DA0:7C24:B6DB:7D52:349F (talk) 15:14, 29 August 2024 (UTC)