- #1
evinda
Gold Member
MHB
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Hello! (Wave)
$$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$
i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$.
Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$?
Do we use the definition, that is the following?
$$R= \frac{1}{\limsup_{n \to +\infty} \sqrt[n]{|a_n|}}$$
If so how can we find the limit $\limsup \sqrt[n]{\frac{1}{n!}}$ ?
$$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$
i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$.
Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$?
Do we use the definition, that is the following?
$$R= \frac{1}{\limsup_{n \to +\infty} \sqrt[n]{|a_n|}}$$
If so how can we find the limit $\limsup \sqrt[n]{\frac{1}{n!}}$ ?