## Homework Statement

Let D be th region in the xy plane in which the series
$$\sum_{k=1}^{\infty}\frac{(x+2y)^k}{k}$$
converges. Describe D.

## The Attempt at a Solution

By the ratio test, we find the radius of converge of the series in x+ 2y to be 1. So, the series will converge when |x+2y| < 1. This region is a rectangle.

What is wrong with this?

Be careful with the ratio test because it only tests for absolute convergence. However you are close. It should be $$x+2y < 1$$ and $$x+2y \geq -1$$. This is not a rectangle. It's a "strip" through the plane.