let-put-S-x-n-0-1-n-n-x-1-prove-that-S-is-C-1-on-0-2-give-the-variation-of-S-x-3-prove-that-x-gt-0-S-x-1-S-x-1-x-4-give-a-equivalent-for-S-at-0-5-find-a-equivalent-for-

Question Number 31533 by abdo imad last updated on 09/Mar/18

l e t p u t S (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n + x}

1) p r o v e t h a t S i s C^{1} o n] 0^{'} + \infty [

2) g i v e t h e v a r i a t i o n o f S (x)

3) p r o v e t h a t \forall x > 0 S (x + 1) + S (x) = \frac{1}{x}

4) g i v e a e q u i v a l e n t f o r S a t 0

5) f i n d a e q u i v a l e n t f o r S a t + \infty .