let-give-I-n-0-1-x-n-1-x-n-dx-1-prove-that-lim-n-gt-I-n-0-2-calculate-I-n-I-n-1-3-find-n-1-1-n-1-n-

Question Number 27666 by abdo imad last updated on 12/Jan/18

l e t g i v e I_{n} = \int_{0}^{1} \frac{x^{n}}{1 + x^{n}} d x

(1) p r o v e t h a t {l i m}_{n - >\propto} I_{n} = 0

(2) c a l c u l a t e I_{n} + I_{n + 1}

(3) f i n d \sum_{n = 1}^{\propto} \frac{{(- 1)}^{n - 1}}{n} .