1. Ch 9.1 Power Series Calculus Graphical, Numerical, Algebraic by
Finney, Demana, Waits, Kennedy

6. 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.

7. For Example Power Series: an infinite sum of variables to a power.
y = sin (x) can be represented as a power series:
Every time you add a term to the series it fits sin x even better.
Let’s check this out using the calculator and then the geometer sketchpad.

14. Power Series for cos x
If sin x can be represented by the power series:
Then cos x can be represented by the power series derived from taking the derivative of sin x:
Let’s check it out on the calculator…

15. Power Series for cos x

16. 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)

17. Sum of Converging Series

18. Power Series Using Calculator

19. Example of a Power Series

23. 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

29. Graph both of these functions on your calculator!

31. Now graph both
y = ln (1+x)
And the power series below and check the fit!

33. Finding Power Series for Functions
Given that 1/(1-x) is represented by the power series:
1 + x + x2 + … + xn + …
On the interval (-1,1),
Find a power series that represent 1/(1+x) on (-1,1).
Find a power series that represents x/(1+x) on (-1,1).
Find a power series that represents 1(1-2x) on (- ½ , ½ )
Find a power series that represents
and give its interval of convergence

34. Finding Power Series for Functions
Given that 1/(1 - x) is represented by the power series:
1 + x + x2 + … + xn + … on the interval (-1,1),
Find a power series that represent 1/(1+x) on (-1,1)
1 - x + x2 – x3 … + (-x)n + …
2. Find a power series that represents x/(1+x) on (-1,1).
x – x2 + x3 – x4 + x5 – x6…. + (-1)n xn+1 + …
3. Find a power series that represents 1/(1 - 2x) on (- ½ , ½ )
1 + 2x + 4x2 + 8x3 + … + (2x)n + …
Find a power series that represents
1 – (x-1) + (x-1)2 – (x-1)3 + … + (-1)n (x-1)n + …
5. Find a power series that represents

35. 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?

36. 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)?
yes
4. Do you think that the series for tan-1 x converges at x = 1? Yes to

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

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