Power Series — Question 3

*Question 3 Find the radius and interval of convergence of $\displaystyle\sum_{n=1}^{\infty}x^n/n$. Identify the harmonic or alternating harmonic series arising at each endpoint.

*Question 3 – Solution

$$\left|\frac{x^{n+1}/(n+1)}{x^n/n}\right|=|x|\frac{n}{n+1}\longrightarrow|x|.$$ The Ratio Test gives convergence for $|x|<1$ and divergence for $|x|>1$, so $R=1$.

The series becomes $\sum1/n$, the divergent harmonic series.

The series becomes $\sum(-1)^n/n$, which converges by the Alternating Series Test (but not absolutely).

The interval is $\boxed{[-1,1)}$. Convergence at the left endpoint is conditional.