Question

How do you find the radius of convergence and interval of convergence for the following?

Sum(n=1 to infinity) ((2x+5)^n) / (n(4^n))

Answer 1

Use the ratio test to determine whether a limit L exists

The ratio of the (n+1)st term : nth term is

|2x+5|/4 * n/(n+1)
as n->∞ that is just |2x+5|/4

the series converges if L < 1, so we need

|2x+5|/4 < 1
|2x+5| < 4
|x + 5/2| < 2

So, the radius of convergence is 2. Now just check the endpoints of the interval

-9/2 <= x <= -1/2

to see whether things converge there.

This article (as well as your text, presumably) should help:

http://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx