Power Series — Question 8

*Question 8 Find the center, radius, and interval of convergence of $$\sum_{n=1}^{\infty}\frac{(-1)^n(x+1)^n}{n}.$$ Track the center carefully and determine which physical endpoint produces the alternating harmonic series.

*Question 8 – Solution

The series is centered at $c=-1$. For $a_n=(-1)^n(x+1)^n/n$, $$\left|\frac{a_{n+1}}{a_n}\right|=|x+1|\frac{n}{n+1}\longrightarrow|x+1|.$$ Thus $|x+1|<1$, so $-2<x<0$ and $R=1$.

Here $x+1=-1$, so $$\frac{(-1)^n(-1)^n}{n}=\frac1n.$$ The harmonic series diverges; $x=-2$ is excluded.

Here $x+1=1$, producing $\sum(-1)^n/n$, which converges conditionally.

The interval is $\boxed{(-2,0]}$.