Timeline for Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 2, 2022 at 6:38 | history | edited | Random Variable | CC BY-SA 4.0 | added 112 characters in body |
| May 2, 2022 at 6:33 | history | edited | Random Variable | CC BY-SA 4.0 | added 112 characters in body |
| Jun 28, 2014 at 16:50 | history | edited | Random Variable | CC BY-SA 3.0 | improved formatting |
| May 11, 2014 at 16:13 | history | bounty awarded | CommunityBot | ||
| May 9, 2014 at 16:44 | comment | added | Random Variable | @robjohn Thanks. | |
| May 9, 2014 at 16:38 | comment | added | robjohn♦ | @user91500: you can pull out the $k=0$ term and handle it separately. In both sums that is $$\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}$$ Then the double sums for $n\ge0$ and $k\ge1$ converge absolutely for $|z|\lt\frac\pi2$. | |
| May 9, 2014 at 15:45 | comment | added | Random Variable | @user91500 The justification might be nontrivial. That's why I didn't post that second evaluation until Peter asked about a different approach. Sufficient, but not necessary, conditions is if $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty}|\frac{2^{2k} z^{2k}}{\pi^{2k}} \frac{(-1)^{n}}{(2n+1)^{2k+1}}| $$ or $$\sum_{k=0}^{\infty} \sum_{n=0}^{\infty} |\frac{2^{2k} z^{2k}}{\pi^{2k}} \frac{(-1)^{n}}{(2n+1)^{2k+1}}|$$converges. The second iterated sum does converge if $k$ starts from $k=1$. That might show that the sum is well-behaved enough to switch the order of summation. Let me ask around. | |
| May 9, 2014 at 14:11 | comment | added | user91500 | $\sum_{n=0}^\infty\sum_{k=0}^\infty|(-1)^n\frac{z^{2k}}{{(2n+1)^{2k+1}}}|$ is divergent, How did you switch the order of summation here? | |
| May 7, 2014 at 4:19 | comment | added | Random Variable | @Peter I added a second proof to my answer. | |
| May 7, 2014 at 4:16 | history | edited | Random Variable | CC BY-SA 3.0 | added 1208 characters in body |
| Apr 30, 2014 at 16:44 | vote | accept | CommunityBot | ||
| Apr 21, 2014 at 22:21 | history | edited | Random Variable | CC BY-SA 3.0 | added 163 characters in body |
| Apr 21, 2014 at 17:33 | history | edited | Random Variable | CC BY-SA 3.0 | added 156 characters in body |
| Apr 21, 2014 at 11:32 | history | edited | Random Variable | CC BY-SA 3.0 | added 79 characters in body |
| Apr 21, 2014 at 11:24 | history | edited | Random Variable | CC BY-SA 3.0 | added 79 characters in body |
| Apr 21, 2014 at 11:15 | history | answered | Random Variable | CC BY-SA 3.0 |