AP Calculus BC 9.1 Power Series, p. 472

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:

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.

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.

Example 1: a r

Example 2: a r

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.)

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.)

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.

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

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.

Example 5: Given: find: hmm?

Example 5:

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