Seria lui Euler zeta(2)=pi^2/6 via integrala din cos
Posted: Sun Mar 09, 2008 1:54 am
Sa se demonstreze ca pentru orice \( n\in\mathbb{N}^{*} \) are loc egalitatea:
\( \int_{0}^{\frac{\pi}{4}}x^2\cos^{2n}xdx=\frac{1\cdot 3\cdot 5\ldots (2n-1)}{2\cdot 4\ldots (2n)}\cdot\frac{\pi}{4}\left(\frac{\pi^2}{6}-\sum_{k=1}^{n}\frac{1}{k^2}\right) \).
Folosind, eventual, rezultatul de mai sus, sa se deduca
\( \lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k^{2}}=\frac{\pi^2}{6} \).
\( \int_{0}^{\frac{\pi}{4}}x^2\cos^{2n}xdx=\frac{1\cdot 3\cdot 5\ldots (2n-1)}{2\cdot 4\ldots (2n)}\cdot\frac{\pi}{4}\left(\frac{\pi^2}{6}-\sum_{k=1}^{n}\frac{1}{k^2}\right) \).
Folosind, eventual, rezultatul de mai sus, sa se deduca
\( \lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k^{2}}=\frac{\pi^2}{6} \).