Difference between revisions of "Aufgaben:Problem 5"
From Ferienserie MMP2
(Created page with "=Problem= Show, by using the Fourier series, that $$\sum_{k=1}^\infty \frac{1}{k^n} = - \frac{(2 \pi i)^n}{2(n!)} B_n,\ n \in 2 \mathbb{Z}_{>0}$$ where \(B_n\) are the Bernoul...") |
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| − | Coming soon ;) [[User:Nik|Nik]] ([[User talk:Nik|talk]]) 18:24, 22 December 2014 (CET) | + | Coming soon ;) |
| + | For the nonce look [http://www.math.ethz.ch/~gruppe5/group5/group5.php?lectures/mmp/hs11/ml03.pdf here] (problem 2) | ||
| + | [[User:Nik|Nik]] ([[User talk:Nik|talk]]) 18:24, 22 December 2014 (CET) | ||
Revision as of 21:17, 22 December 2014
Problem
Show, by using the Fourier series, that $$\sum_{k=1}^\infty \frac{1}{k^n} = - \frac{(2 \pi i)^n}{2(n!)} B_n,\ n \in 2 \mathbb{Z}_{>0}$$ where \(B_n\) are the Bernoulli numbers defined as $$\frac{z}{e^z - 1} = \sum_{n=0}^\infty B_n \frac{z^n}{n!}$$
Hint: consider \( B_n(x) := \sum_{k \in \mathbb{Z} \setminus \{0\}} k^{-n} e^{ikx},\ n \in \mathbb{Z}_{\geq 2},\ x \in \mathbb{R} \)
Solution
Coming soon ;) For the nonce look here (problem 2) Nik (talk) 18:24, 22 December 2014 (CET)
