Are you familiar with the ratio test?. That is what we usually use to test convergences.

Anyway:

When we apply the ratio test, we can note that

The ratio test implies the series converges if |x|<1 and diverges if

|x|>1.

The ratio test does not provide info for |x|=1 or |x|=-1, so we can test

them separately by subbing into the original series.

x=1:

This is a conditionally convergent harmonic series.

For x=-1, we get:

This is a divergent harmonic series.

Therefore, the interval of convergence for the series is (-1,1] and the radius of convergence is R=1.