# Radius of convergence for the power series

I have a feeling I am making a very silly mistake here but is the limit for $\lim_{n \to \infty} |\frac {zn^n}{(n+1)^{n+1}}| =|z|$ ?

just need a confirmation.

Thanks guys.

## 2 Answers

Seems to work like this: $$\lim_{n\to\infty} \bigg|\frac{zn^n}{(n+1)^{n+1}}\bigg|=\lvert z\rvert\lim_{n\to\infty}\bigg|\frac{n^n}{(n+1)^{n+1}}\bigg|=\lvert z\rvert\lim_{n\to\infty}\bigg|\frac{1}{n}\bigg|=0.$$ The third equality follows from expanding the denominator. We can see that the dominating term is the term with power $(n+1)$: that is $n^{n+1}$. If we divide the numerator and denominator by $n^n$ and take the limit, we get the result.

• Or observe that $n^n/(n+1)^{n+1}=[(n/(n+1))^n]/[n+1]<1/[n+1].$.........+1 – DanielWainfleet Feb 3 '17 at 3:20
• That works too! – Antonios-Alexandros Robotis Feb 3 '17 at 3:21

You should know that $\lim\limits_{n\to\infty} \left( \dfrac n {n+1}\right)^n = \dfrac 1 e,$ and what you've got is $\lim\limits_{n\to\infty} \left( \dfrac n {n+1} \right)^n \cdot \dfrac 1 {n+1}.$