Find out the answer to "How can I calculate∑n=1n=1(−1)n16n2−1using the Fourier series?"

iristh3virusoo2

Answered question

2022-02-24

How can I calculate

\sum_{n = 1}^{n = 1} \frac{{(- 1)}^{n}}{16 n^{2} - 1}

using the Fourier series?

Elena Ray

Beginner2022-02-25Added 4 answers

| \sin (x) | = \frac{2}{π} - \frac{4}{π} \sum_{n = 1}^{\infty} \frac{\cos (2 n x)}{4 n^{2} - 1}

∴ | \sin (\frac{π}{4}) | = \frac{2}{π} - \frac{4}{π} \sum_{n = 1}^{\infty} \frac{\cos (\frac{π x}{2})}{4 n^{2} - 1}

= \frac{2}{π} - \frac{4}{π} \sum_{n = even}^{\infty} \frac{{(- 1)}^{\frac{n}{2}}}{4 n^{2} - 1}

(n \to 2 m) = \frac{2}{π} - \frac{4}{π} \sum_{m = 1}^{\infty} \frac{{(- 1)}^{m}}{16 m^{2} - 1}

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{16 n^{2} - 1} = \frac{\frac{2}{π} - \frac{\sqrt{2}}{2}}{\frac{4}{π}}

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