Calculating the following series: $\sum_{k=0}^{\infty}(−1)^k2k+1$

Question

Here are the series:

$$ \sum_{k=0}^{\infty}\frac{(−1)^k}{2k+1} $$

I don't know how to start because obviously can't compute all the values up to infinity. My only thought is to use d'alembert ratio? but doesn't this only tell me if the series converges? not what the actual sum of the series is? Thank you for your help.

See https://math.stackexchange.com/questions/2227555/compute-sum-limits-n-0-infty-frac12n1-4n — lab bhattacharjee, Apr 19 '17 at 08:53
Why are they both the same series? I fixed it so it looks nicer but I'm still confused. — Harambe, Apr 19 '17 at 08:53
hey! sorry not sure ho to write mathematically on here. yes Italy 4 that is right except should be (-1)^k / 2k +1 — Kate, Apr 19 '17 at 08:54

score 1 · Answer 1 · 2017-04-19T09:16:37.457

1

Hint:

By summation of the geometric series,

$$\sum_{k=0}^\infty(-1)^kx^{2k}=\frac1{1+x^2}$$ and integrating term-wise over $[0,x)$,

$$\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{2k+1}=\arctan x.$$

edited Apr 19 '17 at 09:16

answered Apr 19 '17 at 09:01

Calculating the following series: $\sum_{k=0}^{\infty}(−1)^k2k+1$

1 Answers1