Calculus 9.4

1
9.4 Radius of Convergence
Greg Kelly, Hanford High School, Richland,
Washington
2
Convergence
The series that are of the most interest to us
are those that converge.
Today we will consider the question
3
The first requirement of convergence is that the
terms must approach zero.
Ex. 2
Note that this can prove that a series diverges,
but can not prove that a series converges.
4
There are three possibilities for power series
convergence.
The series converges over some finite
interval (the interval of convergence).
The series may or may not converge at the
endpoints of the interval.
(As in the previous example.)
The number R is the radius of convergence.
5
Direct Comparison Test
This series converges.
For non-negative series
So this series must also converge.
If every term of a series is less than the
corresponding term of a convergent series, then
both series converge.
So this series must also diverge.
If every term of a series is greater than the
corresponding term of a divergent series, then
both series diverge.
This series diverges.
6
Ex. 3
Prove that converges for all real
x.
There are no negative terms
larger denominator
The direct comparison test only works when the
terms are non-negative.
7
Absolute Convergence
The term converges absolutely means that the
series formed by taking the absolute value of
each term converges. Sometimes in the English
language we use the word absolutely to mean
really or actually. This is not the case
here!
If the series formed by taking the absolute value
of each term converges, then the original series
must also converge.
If a series converges absolutely, then it
converges.
8
Ex. 4
We test for absolute convergence
9
Ratio Technique
We have learned that the partial sum of a
geometric series is given by
10
Geometric series have a constant ratio between
terms. Other series have ratios that are not
constant. If the absolute value of the limit of
the ratio between consecutive terms is less than
one, then the series will converge.
11
Ex
If we replace x with x-1, we get
If the limit of the ratio between consecutive
terms is less than one, then the series will
converge.
12
If the limit of the ratio between consecutive
terms is less than one, then the series will
converge.
13
Ex
14
Ex
15
Ex
16
Ex
Note If R is infinite, then the series converges
for all values of x.
17
Another series for which it is easy to find the
sum is the telescoping series.
Using partial fractions
Ex. 6
p