1. 9.1 Power Series

2. e 2

3. What You Will Learn
All continuous functions can be represented as a polynomial
Polynomials are easy to integrate and differentiate
Calculators use polynomials to calculate trig functions, logarithmic functions etc.
Downfall of polynomial equivalent functions is that they have an infinite number of terms.

4. 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:

5. 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.

6. 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.

7. Geometric Series Partial Sum of a Geometric Series:
Sn = a + ar + ar2 + ar3 + … + arn-1
-[r Sn = ar + ar2 + ar3 + … + arn
Sn – r Sn = a + arn
Sn (1 – r) = a (1 - rn)

8. Sum of Converging Series

9. Power Series Using Calculator

10. Example 1: a r

11. Example 2: a r

12. 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.)

13. Example of a Power Series

14. A power series is in this form:
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.)

15. 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.

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

17. Example 5:
Given:
find:
hmm?

18. Example 5:

19. 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

20. Convergent Series Only two kinds of series converge:
1) Geometric whose | r | < 1
2) Telescoping series
Example of a telescoping series: the middle terms cancel out

21. Finding a series for tan-1 x
1. Find a power series that represents on (-1,1)
Use integration to find a power series that represents
tan-1 x.
Graph the first four partial sums. Do the graphs suggest convergence on the open interval (-1, 1)?
4. Do you think that the series for tan-1 x converges at x = 1?

22. Guess the function Define a function f by a power series as follows:
Find f ‘(x).
What function is this?