Calculus Methods
30 Intervals and Radii of Convergence
After looking at which series converge and which don't, it is
finally time to use them. When series are not simply series of
constants, but also of variables, life gets interesting.
1) Identify the value
about which the series is centered (the 
in
your power series, when phrased in the
form 
.
2) Find the Radius of Convergence (see lesson 55).
2A) You will usually use the Root Test (see lesson 53), since
you are dealing with powers of x.
2B) Don't forget to put your Root Test (or Ratio Test) answer
into the form 
. 
is
your Radius.
3) The Interval of
Convergence will be from 
to
, but you
need to check those endpoints.
4) Check the endpoints
by plugging in the values 
and
into the function. If the series
converges at that point,
include the endpoint; if not, then do not include the endpoint.
Example #1: Find the
Interval of Convergence of 
.
1) We are centered
about 
.
2) Using the Root
Test, we find 
, so 
.
3) Our interval then, will be from 1 to 3.
4)
, which is an alternating series that
does
not converge (fails the nth term test).
, which also diverges due to the nth term
test.
So we do not include either endpoint, and our Interval of
Convergence is 
.
Example #2: Find the
Interval of convergence of 
.
1) We are centered
about 
.
2)
The "problem" here is that the left side NEVER converges. It
always approaches infinity. Thus, there are no values for
which this function converges and 
.
3) There is no Radius of Convergence, so there are no endpoints.
4) No Interval of Convergence.
Example #3: Find the Interval of Convergence of the function
.
1) We are centered
about 
2)
Here we have an example where the series ALWAYS
converges, so the Radius is infinite.
3) There are no endpoints since the radius is infinite.
4) The Interval of
Convergence is 
.
Example #4: Find the Interval of Convergence of the function
.
1) We are centered
about 
.
2)
Our Radius is 
3) Our endpoints are -12 and -2.
4)
, which is the Alternating
Harmonic Series, which converges.
, which is the Harmonic Series,
which diverges.
Our interval of
convergence is thus 
.
On to Method 31 - Generating Geometric Power Series