Sum 40090

Calculus Level 3

Compute

$\large \int_0^1 \left(\sum _{n=2}^{\infty } \frac{\zeta (n) \cos (\pi n x)}{n}\right) \, dx$

Notation: $\zeta(\cdot)$ denotes the Riemann zeta function .

The answer is 0.

1 solution

Chew-Seong Cheong
Dec 26, 2017

$\begin{aligned} I & = \int_0^1 \sum_{n=2}^\infty \frac {\zeta(n) \cos (\pi nx)}n dx \\ & = \sum_{n=2}^\infty \frac {\zeta(n)}n \int_0^1 \cos (\pi nx) \ dx \\ & = \sum_{n=2}^\infty \frac {\zeta(n)}n \left[\frac {\sin (\pi nx)}{\pi n} \right]_0^1 \\ & = \boxed{0} \end{aligned}$

Would you mind showing how to apply the dominated convergence theorem to this problem (assuming you can)?

James Wilson - 5 months, 1 week ago

Sum 40090

The answer is 0.

1 solution

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