let-u-n-1-n-0-pi-2-sin-n-xdx-calculate-u-n-

Question Number 52675 by maxmathsup by imad last updated on 11/Jan/19

l e t u_{n} = {(- 1)}^{n} \int_{0}^{\frac{π}{2}} {s i n}^{n} x d x c a l c u l a t e Σ u_{n}

Commented by maxmathsup by imad last updated on 11/Jan/19

w e h a v e \sum_{n = 0}^{\infty} u_{n} = \sum_{n = 0}^{\infty} \int_{0}^{\frac{π}{2}} {(- s i n x)}^{n} d x = \int_{0}^{\frac{π}{2}} (\sum_{n = 0}^{\infty} {(- s i n x)}^{n}) d x

= \int_{0}^{\frac{π}{2}} \frac{d x}{1 + s i n x} =_{t a n (\frac{x}{2}) = t} \int_{0}^{1} \frac{1}{1 + \frac{2 t}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}} = 2 \int_{0}^{1} \frac{d t}{1 + t^{2} + 2 t}

= 2 \int_{0}^{1} \frac{d t}{{(t + 1)}^{2}} = 2 {[- \frac{1}{t + 1}]}_{0}^{1} = 2 (1 - \frac{1}{2}) = 1 \Rightarrow \sum_{n = 0}^{\infty} u_{n} = 1 .