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Contents Index Dark Mode Prev Up Next \(\def\versionnumber{1.54}
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Section 10.1 Infinite Sums
In this exercise, we evaluate the sums
\(\sum_{k \geq 1} \frac 1 {k^2}\) and
\(\sum_{k \geq 1} \frac{
(-1)^k }{k^2}\text{.}\) We hope the idea of how to compute such sums in general will become clear.
Consider the function
\(f(z) = \frac{ \pi \cot (\pi z) }{ z^2
}\text{.}\) Compute the residues at all the singularities of
\(f\text{.}\)
Let
\(N\) be a positive integer and
\(\gamma_N\) be the rectangular path from
\(N+\frac 1 2
-iN\) to
\(N+\frac 1 2 +iN\) to
\(-N-\frac 1 2 +iN\) to
\(-N-\frac 1 2 -iN\) back to
\(N+\frac 1 2 -iN\text{.}\)
Show that
\(| \cot (\pi z) | \lt 2\) for
\(z \in \gamma_N\text{.}\) (
Hint : Use
Exercise 3.6.35 .)
Show that
\(\lim_{N \to \infty} \int_{\gamma_N} f = 0\text{.}\)
Use the Residue
Theorem 9.2.2 to arrive at an identity for
\(\sum_{k \in \Z \setminus
\{0\}} \frac 1 {k^2}\text{.}\)
Evaluate
\(\sum_{k \geq 1} \frac 1 {k^2}\text{.}\)
Repeat the exercise with the function \(f(z) = \frac{ \pi }{ z^2 \sin (\pi z) }\) to arrive at an evaluation of
\begin{equation*}
\sum_{k \geq 1} \frac{ (-1)^k }{k^2} \,\text{.}
\end{equation*}
(Hint : To bound this function, you may use the fact that \(\frac 1 {\sin^2(z)} = 1 +
\cot^2(z)\text{.}\) )
Evaluate
\(\sum_{k \geq 1} \frac 1 {k^4}\) and
\(\sum_{k
\geq 1} \frac{ (-1)^k }{k^4}\text{.}\)
We remark that, in the language of
Example 7.2.17 , you have computed the evaluations
\(\zeta(2)\) and
\(\zeta(4)\) of the Riemann zeta function. The function
\(\zeta^*(z) := \sum_{k \geq 1} \frac{ (-1)^k
}{k^z}\) is called the
alternating zeta function .
