**1. Definitions.** Let be a variable. a series of the form

(1)

is called a **“Power Series about ”**.

A series of the form

(2)

is called a **“Power Series about ”**.

Example 1.Consider the series

(3)

For what values of does this series converge? When will the series diverge?

Solution.Using the absolute ratio test, we see

(4)

For convergence, we need

(5)

So divergence would be when

(6)

We need to check the case. Observe, for , we have

(7)

which diverges. And at we have

(8)

which still diverges!

**2. Theorem.** Let and

(9)

converge at . Then converges absolutely for .

*Proof:* Let be any value such that . Since converges, there exists an such that

(10)

Well, we see

(11)

so

(12)

and moreover

(13)

But observe the series

(14)

is a geometric series which converges since .

Therefre the series converges by comparison, and the absolute convergence test tells us converges. Therefore is absolutely convergent. ▮

Example 2.Find the values of for which

(15)

converge.

Solution: Using the absolute ratio test, we find

(16)

We get convergence for . So

(17)

We have to check the boundary cases.

When , we have

(18)

which diverges by the integral test (or the comparison test with the Harmonic series). For we have our series become

(19)

which converges by the alternating series test.

**Exercise 1.** Find the values of for which

converges.