DEFINITION: If are constants
and is a variable, then a series of the form

is called a power series in and a series of
the form

is called a power series in

EXAMPLES:

1.

2.

3.

4.

5.

THEOREM: For any power series exactly one of
the following is true:

(a) The series converges only for

(b) The series converges absolutely (and hence converges)
for all real values of

(c) The series converges absolutely (and hence converges)
for all in some finite open interval and diverges if
or At either of the
values or the series may converge absolutely, converge
conditionally, or diverge, depending on the particular
series.

THEOREM: For a power series exactly one of
the following is true:

(a) The series converges only for

(b) The series converges absolutely (and hence converges)
for all real values of

(c) The series converges absolutely (and hence converges)
for all in some finite open interval and diverges if
or At either of the
values or the series may converge absolutely, converge
conditionally, or diverge, depending on the particular
series.

EXAMPLE: Find the interval of convergence of the series

Solution: By the ratio test we have

Thus, the series converges absolutely if

The series diverges if or

Now we examine the endpoints. If then

which converges by the alternating series test. Moreover,
it converges absolutely (see below). Similarly, if then

which converges by the -series test.
So, the interval of convergence is and the radius
of convergence is

EXAMPLE: Find the interval of convergence of the series

Solution: By the ratio test we have

Thus, the series converges absolutely if

The series diverges if or

Now we examine the endpoints. If then

which converges by the alternating series test. Moreover,
it converges conditionally (see below). Similarly, if then

which diverges by the -series test.
So, the interval of convergence is and the radius
of convergence is

EXAMPLE: Find the interval of convergence of the series

Solution: By the ratio test we have

Thus, the series converges absolutely if

The series diverges if or

Now we examine the endpoints. If then

This series is divergent by the divergence test since . Similarly,
if then

This series is also divergent by the divergence test,
since doesn't exist. So, the interval of
convergence is and the radius of convergence is

EXAMPLE: Find the interval of convergence of the series

Solution: By the ratio test we have

Thus, the series converges absolutely if

The series diverges if or Now we
examine the endpoints. If then

This series is convergent by the alternating series test.
Moreover, it converges conditionally (see below).
Similarly, if then

This series is divergent by the -test. So, the
interval of convergence is and the
radius of convergence is

EXAMPLE: Find the interval of convergence of the series

Solution: By the ratio test we have

Since it follows that the series converges if

and diverges otherwise. The radius of convergence is

EXAMPLE: Find the interval of convergence of the series

Solution 1: By the root test we have

Since regardless of the value of this power
series will converge absolutely for every . In these
cases we say that the radius of convergence is and interval
of convergence the whole number line.

Solution 2: By the ratio test we have

and the same result follows.

EXAMPLE: Find the interval of convergence of the series

Solution 1: By the root test we have

Thus, the series converges absolutely if

The series diverges if or Now we examine
the endpoints. If then

This series is divergent by the divergence test, since doesn't
exist. So, the interval of convergence is and the radius
of convergence is