1. AP Calculus BC
9.1 Power Series, p. 472

2. Start with a square one unit by one unit:1
This is an example of an infinite series. 1
This series converges (approaches a limiting value.)
Many series do not converge:

3. a1, a2,… are terms of the series. an is the nth term.
In an infinite series:
a1, a2,… are terms of the series.
an is the nth term.
Partial sums:
nth partial sum
If Sn has a limit as , then the series converges, otherwise it diverges.
Or if then the series converges, otherwise it diverges.

4. Geometric Series:
In a geometric series, each term is found by multiplying the preceding term by the same number, r.
This converges to if , and diverges if
is the interval of convergence.

5. Example 1: a r

6. Example 2: a r

7. The partial sum of a geometric series is:
If then
If and we let , then:
The more terms we use, the better our approximation (over the interval of convergence.)

8. is a power series centered at x = 0.
A power series is in this form:
is a power series centered at x = 0. or
The coefficients c0, c1, c2… are constants.
The center “a” is also a constant.
(The first series would be centered at the origin if you graphed it. The second series would be shifted left or right. “a” is the new center.)

9. Once we have a series that we know, we can find a new series by doing the same thing to the left and right hand sides of the equation.
Example 3:
This is a geometric series where r = −x.
To find a series for
multiply both sides by x.

10. Example 4:
Given:
find:
So:
We differentiated term by term.

11. Example 4b:
First, note 2nxn can be written as (2x)n.
Taking the derivative of both sides yields:
Note n starts at 1 now instead of 0.

12. Example 5:
Given:
find:
hmm?

13. Example 5:

14. The previous examples of infinite series approximated simple functions such as or .
This series would allow us to calculate a transcendental function to as much accuracy as we like using only pencil and paper! p