Question Number 36748 by prof Abdo imad last updated on 05/Jun/18
![let f(x)= Σ_(n=1) ^∞ (((−1)^(n−1) )/(ln(nx))) 1) give D_f and study f on]1,+∞[ 2)study the continjity of f and calculate lim _(x→1) f(x) and lim_(x→+∞) f(x). 3) prove that f is C^1 on ]1,+∞[ .](https://www.tinkutara.com/question/Q36748.png)
$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{ln}\left({nx}\right)} \\ $$$$\left.\mathrm{1}\left.\right)\:{give}\:{D}_{{f}} \:\:{and}\:{study}\:{f}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continjity}\:{of}\:{f}\:{and}\:{calculate} \\ $$$${lim}\:_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]\mathrm{1},+\infty\left[\:.\right. \\ $$