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AP Calculus BC 9.1 Power Series, p. 472.

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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


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