$$\newcommand{\mtn}{\mathbb{N}}\newcommand{\mtns}{\mathbb{N}^*}\newcommand{\mtz}{\mathbb{Z}}\newcommand{\mtr}{\mathbb{R}}\newcommand{\mtk}{\mathbb{K}}\newcommand{\mtq}{\mathbb{Q}}\newcommand{\mtc}{\mathbb{C}}\newcommand{\mch}{\mathcal{H}}\newcommand{\mcp}{\mathcal{P}}\newcommand{\mcb}{\mathcal{B}}\newcommand{\mcl}{\mathcal{L}} \newcommand{\mcm}{\mathcal{M}}\newcommand{\mcc}{\mathcal{C}} \newcommand{\mcmn}{\mathcal{M}}\newcommand{\mcmnr}{\mathcal{M}_n(\mtr)} \newcommand{\mcmnk}{\mathcal{M}_n(\mtk)}\newcommand{\mcsn}{\mathcal{S}_n} \newcommand{\mcs}{\mathcal{S}}\newcommand{\mcd}{\mathcal{D}} \newcommand{\mcsns}{\mathcal{S}_n^{++}}\newcommand{\glnk}{GL_n(\mtk)} \newcommand{\mnr}{\mathcal{M}_n(\mtr)}\DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\sh}{sh}\DeclareMathOperator{\th}{th} \DeclareMathOperator{\vect}{vect}\DeclareMathOperator{\card}{card} \DeclareMathOperator{\comat}{comat}\DeclareMathOperator{\imv}{Im} \DeclareMathOperator{\rang}{rg}\DeclareMathOperator{\Fr}{Fr} \DeclareMathOperator{\diam}{diam}\DeclareMathOperator{\supp}{supp} \newcommand{\veps}{\varepsilon}\newcommand{\mcu}{\mathcal{U}} \newcommand{\mcun}{\mcu_n}\newcommand{\dis}{\displaystyle} \newcommand{\croouv}{[\![}\newcommand{\crofer}{]\!]} \newcommand{\rab}{\mathcal{R}(a,b)}\newcommand{\pss}[2]{\langle #1,#2\rangle} $$
Bibm@th

Rayon de convergence - 2 - Bibm@th.net

Exercice 1 - Rayon de convergence - 2 [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Déterminer le rayon de convergence des séries entières suivantes : $$\begin{array}{lll} \mathbf{1.}\ \sum_n\frac{(1+i)^n z^{3n}}{n\cdot 2^n}& \mathbf{2.}\ \sum_{n\geq 1}\ln\left(1+\sin\frac1n\right)x^n& \mathbf{3.}\ \sum_{n\geq 1}\big(\exp(1/n)-1\big)x^n\\ \mathbf{4.}\ \sum_n a^{\sqrt{n}}z^n,\textrm{ }a>0& \mathbf{5.}\ \sum_n z^{n!}&\mathbf{6.}\ \sum_n n^{\ln n}z^n\\ \end{array}$$
Indication
Corrigé