Power Series

1
Power Series

• Radii and Intervals of Convergence

2
First some examples
Consider the following example series

• What does our intuition tell us about the
  convergence or divergence of this series?
• What test should we use to confirm our intuition?

3
Power Series
Now we consider a whole family of similar series

• What about the convergence or divergence of
  these series?
• What test should we use to confirm our intuition?

We should use the ratio test furthermore, we
can use the similarity between the series to test
them all at once.
4
We start by setting up the appropriate limit.
How does it go?
5
What are Power Series?
Its convenient to think of a power series as an
infinite polynomial
Polynomials
Power Series
6
In general. . .
Definition A power series is a (family of)
series of the form
In this case, we say that the power series is
based at x0 or that it is centered at x0.
What can we say about convergence of power
series? A great deal, actually.
7
Checking for Convergence
8
Checking for Convergence
Checking on the convergence of
We start by setting up the appropriate limit.
The ratio test says that the series converges
provided that this limit is less than 1. That
is, when xlt1.
9
What about the convergence of
We start by setting up the ratio test limit.
Since the limit is 0 (which is less than 1), the
ratio test says that the series converges
absolutely for all x.
10
Now you work out the convergence of
Dont forget those absolute values!
11
Now you work out the convergence of
We start by setting up the ratio test limit.
What does this tell us?

• The power series converges absolutely when
  x3lt1.
• The power series diverges when x3gt1.
• The ratio test is inconclusive for x -4 and x
  -2. (Test these separately what happens?)

12
Convergence of Power Series
What patterns can we see? What conclusions can
we draw?
When we apply the ratio test, the limit will
always be either 0 or some positive number times
x-x0. (Actually, it could be ?, too. What
would this mean?)

• If the limit is 0, the ratio test tells us that
  the power series converges absolutely for all x.
• If the limit is kx-x0, the ratio test tells us
  that the series converges absolutely when
  kx-x0lt1. It diverges when kx-x0gt1. It fails
  to tell us anything if kx-x01.

13
Suppose that the limit given by the ratio test is

• We need to consider separately the cases when
• k x-x0 lt 1 (the ratio test guarantees
  convergence),
• k x-x0 gt 1 (the ratio test guarantees
  divergence), and
• k x-x0 1 (the ratio test is inconclusive).
• This means that . . .

Recall that k ? 0 !
14
Recapping
15
Conclusions

• Theorem If we have a power series
  ,
• It may converge only at xx0.

• It may converge for all x.
• It may converge on a finite interval centered at
  xx0.

16
Conclusions

• Theorem If we have a power series
  ,
• It may converge only at xx0.

• It may converge for all x.
• It may converge on a finite interval centered at
  xx0.