10 1 Power Series Photo by Vickie Kelly

10. 1 Power Series Photo by Vickie Kelly, 2003 Jkoury 2017

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. )

Here is another way to demonstrate that this series converges: Let: Multiply both sides by ½. Subtract the result from the original series.

Here is another way to demonstrate that this series converges: Let: Subtract the result from the original series. Many series do not converge:

In an infinite series: a 1, a 2, … are terms of the series. an is the nth term. Partial sums: nth partial sum If Sn has a limit as otherwise it diverges. , then the series converges,

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. )

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. )

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 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