Power Series — Question 3

Question 3 – Solution

Step 1: Find the interior interval.

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

Step 2: Test $x=1$.

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

Step 3: Test $x=-1$.

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

Conclusion.

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