9.1 Power Series AP Calculus BC

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

In an infinite series: a 1, a 2,… are terms of the series. a n is the n th term. Partial sums: n th partial sum If S n 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

a r Example 2:

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.)

A power series is in this form: or The coefficients c 0, c 1, c 2 … 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.) is a power series centered at x = 0.

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. This is a geometric series where r = −x. To find a series for multiply both sides by x. Example 3:

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! 