Question Number 31533 by abdo imad last updated on 09/Mar/18
![let put S(x)=Σ_(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>0 S(x+1)+S(x)=(1/x) 4)give a equivalent for S at 0 5)find a equivalent for S at +∞.](https://www.tinkutara.com/question/Q31533.png)
$${let}\:{put}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+{x}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{S}\:{is}\:{C}^{\mathrm{1}} \:{on}\right]\mathrm{0}'+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){give}\:{the}\:{variation}\:{of}\:{S}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall{x}>\mathrm{0}\:{S}\left({x}+\mathrm{1}\right)+{S}\left({x}\right)=\frac{\mathrm{1}}{{x}} \\ $$$$\left.\mathrm{4}\right){give}\:{a}\:{equivalent}\:{for}\:{S}\:{at}\:\mathrm{0} \\ $$$$\left.\mathrm{5}\right){find}\:{a}\:{equivalent}\:{for}\:{S}\:{at}\:+\infty. \\ $$