9 9 Power Series A power series is

9. 9 Power Series

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

Example 5

Example 5

Example 3

Example 3

Example 2

Example 2

Series that we should know

We have saved the best for last!

An amazing use for infinite series: Substitute xi for x. Factor out the i terms.

This is the series for cosine. This is the series for sine. Let This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated. p

p. 676 5 - 25 odd • When a mathematician writes a Fantasy book, will the page numbers be imaginary numbers?