Radius of convergence: $\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$

Question

It is asked to find the radius of convergence of the series

$$\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$$

i.e, to find the values of x such that this series converges.

Clearly, I could directly apply the root test and find the interval of convergence of the series. But, my doubt is if I can consider this a power series and evaluate

$$\lim_{k \to \infty}\left\{|\frac{1}{2k-1}|\right\}^{\frac{1}{2k-1}}$$

Is that correct? Or should we do

$\lim_{k \to \infty}\left\{\left|\frac{1}{2k-1}\right|\right\}^{\frac{1}{k}}?$

Thanks in advance!

Answer 1

The root test is the correct idea here. To see which root you need to take, though, keep in mind that you could 'refill' your sum with the missing powers of $x$ by filling in a coefficient of $0$, thus we have $\sum_{k=1}^{\infty}{\frac{x^{2k-1}}{2k-1}}=\sum_{k=1}^{\infty}{x^na_n}$, where $a_k=\frac{1}{k}$ for odd $k$ and $a_k=0$ for even $k$.

Now the root test tells us to find $\limsup \sqrt[n]{|a_n|}$, where $\sqrt[n]{|a_n|}$ is $0$ for some $n$, and $\sqrt[n]{\frac{1}{n}}$ for the others.

This leaves you with a simple calculation to get the radius of convergence; the two values you still need to try manually aren't much work, either.

Can you take it from here?