interval of convergence of the power series $\sum_{n=1}^\infty(1+\frac{1}{n})^{n^2}x^n$

Question

Find the radius of convergence and the interval of convergence of the power series

$\sum_{n=1}^\infty(1+\frac{1}{n})^{n^2}x^n$

My attempt: The root test brings us to

$\lim_{n\rightarrow\infty}|(1+\frac{1}{n})^nx|$,

but I'm not sure where this brings me. Is the ratio test any help?

Any help appreciated!

Answer 1

Note: Both tests, root test as well as ratio test can be used to analyze the convergence behavior. But, the first one is in this case considerably simpler to use.

We recall the validity of the limit \begin{align*} \lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^n=e \end{align*}

Root test:

We obtain \begin{align*} \lim_{n\rightarrow\infty}\left|\sqrt[n]{\left(1+\frac{1}{n}\right)^{n^2}x^n}\right| =|x|\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} =|x|e \end{align*}

and we conclude the series is convergent for \begin{align*} |x|<\frac{1}{e} \end{align*} and divergent for $|x|>\frac{1}{e}$.

$$ $$

Ratio test:

We obtain \begin{align*} \lim_{n\rightarrow\infty}\left|\frac{\left(1+\frac{1}{n+1}\right)^{(n+1)^2}x^{n+1}} {\left(1+\frac{1}{n}\right)^{n^2}x^n}\right| &=|x| \lim_{n\rightarrow\infty}\frac{\left(1+\frac{1}{n+1}\right)^{n^2+2n+1}} {\left(1+\frac{1}{n}\right)^{n^2}}\\ &=|x|\lim_{n\rightarrow\infty}\left(\frac{n(n+2)}{(n+1)(n+1)}\right)^{n^2} \left(1+\frac{1}{n+1}\right)^{2n+1}\\ &=|x|\lim_{n\rightarrow\infty} \underbrace{\frac{1}{\left(1+\frac{1}{n^2+2n}\right)^{n^2}}}_{\longrightarrow \ \ \frac{1}{e}} \underbrace{\left(1+\frac{1}{n+1}\right)^{2n}}_{\longrightarrow\ \ e^2} \underbrace{\left(1+\frac{1}{n+1}\right)^{1}}_{\longrightarrow \ \ 1}\\ &=|x|e \end{align*} and we conclude as above.