Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$
Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I wasn't able to find this using the root test however. Is there a clever way of finding this with the root test?
Best Answer
$$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+1}{n}} = \lim_{n \to \infty}\sqrt[n]{\frac{2^n}{n}} =\lim_{n \to \infty}\frac{2}{ \sqrt[n]{n}}=2$$
Why can we omit the $+1$ term? Good question. ;)
Write $$2^n+1 = 2^n(1+\frac{1}{2^n})$$
As $n \to \infty$, the limit of both sides becomes equal. Therefore $$\lim_{n \to \infty} 2^n+1 = \lim_{n \to \infty} 2^n(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n\cdot \lim_{n\to \infty}(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n \cdot 1= \lim_{n \to \infty}2^n$$