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Question Number 165240 by SANOGO last updated on 28/Jan/22

 soit la serie de fonction Σ_(n=2   ) (x^n /(nx+ln(n)))  etudie la convergence simple sur [0,1[

$$\:{soit}\:{la}\:{serie}\:{de}\:{fonction}\:\underset{{n}=\mathrm{2}\:\:\:} {\sum}\frac{{x}^{{n}} }{{nx}+{ln}\left({n}\right)} \\ $$$${etudie}\:{la}\:{convergence}\:{simple}\:{sur}\:\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$

Answered by mindispower last updated on 28/Jan/22

∀a∈[0,1]  0≤Σ_(n≥2) (a^n /(na+ln(n)))≤Σ_(n≥2) (a^n /(ln(2)))  Σ_(n≥2) (a^n /(ln(2))).cv ∀a∈[0,1[  ⇒Σ_(n≥2) (x^n /(nx+ln(2))) Cv

$$\forall{a}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\mathrm{0}\leqslant\underset{{n}\geqslant\mathrm{2}} {\sum}\frac{{a}^{{n}} }{{na}+{ln}\left({n}\right)}\leqslant\underset{{n}\geqslant\mathrm{2}} {\sum}\frac{{a}^{{n}} }{{ln}\left(\mathrm{2}\right)} \\ $$$$\underset{{n}\geqslant\mathrm{2}} {\sum}\frac{{a}^{{n}} }{{ln}\left(\mathrm{2}\right)}.{cv}\:\forall{a}\in\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$$$\Rightarrow\underset{{n}\geqslant\mathrm{2}} {\sum}\frac{{x}^{{n}} }{{nx}+{ln}\left(\mathrm{2}\right)}\:{Cv} \\ $$

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