We want to find all numbers for which the power series

(1)

converges. Define for every and use Ratio test to write

(2)

We know if then the infinite sum is finite by Ratio test and when it is divergent. However, Ratio test fails when . We check these here. When we get

(3)

and the divergent test tells us

(4)

Therefore is not acceptable. Next when we get

(5)

Again the divergence test tells us

(6)

Therefore only when the given power series converges. Therefore the interval of convergence is with center and radius of convergence equal to

Suppose we are given the power series

(7)

We would like to determine for what values of this power series is convergent. Use Ratio test to write

(8)

We know when

(9)

the given power series is convergent that is

(10)

We know when , the power series is divergent but what about when ? The Ratio test fails here. We check the endpoints of the interval of convergence and When we get

(11)

Since , the divergence test tells us the power series is divergent. Next, when we get

(12)

We have shown in the past that the series is divergent. So, the interval of convergence is center at and radius of convergence equal to

We want to use geometric series to write as a power series. Differentiate and write

(13)

We know, when

(14)

Now integrate over the interval

(15)

On the other hand we know

(16)

Therefore

(17)

Let us assume that has a Taylor expansion at and we would like to construct this. Write

(18)

We know if has a Taylor expansion at then

(19)

For this example, and so

(20)