Power Series — Question 4

*Question 4 Find the radius and interval of convergence of $\displaystyle\sum_{n=1}^{\infty}x^n/n^2$. Explain why its radius matches Question 3 while both endpoints now converge.

*Question 4 – Solution

$$\left|\frac{x^{n+1}/(n+1)^2}{x^n/n^2}\right|=|x|\left(\frac{n}{n+1}\right)^2\longrightarrow|x|.$$ Thus the interior condition is $|x|<1$ and $R=1$.

• At $x=1$, the series is $\sum1/n^2$, which converges.

• At $x=-1$, the absolute-value series is again $\sum1/n^2$, so the series converges absolutely.

The interval is $\boxed{[-1,1]}$. Both this series and $\sum x^n/n$ have coefficient roots tending to $1$, hence the same radius, but the stronger $n^{-2}$ decay makes both boundary series absolutely convergent.