13.8 Power Series

**Disclaimer!! Do not get bogged down by notation, it is just series!** A polynomial is a relatively simple function to calculate. They are used in a power series to evaluate a sometimes more complicated function. A power series is simply the sum of a sequence with increasing powers of x. *notice k starts at 0 to allow a general term *also notice the coefficients are not necessarily the same, if they are, we have an infinite geometric series.

geometric a 1 = 2 and r = x since it’s geometric, it will converge if  r  =  x  < 1  –1 < x < 1 it will also converge to  x  < 1  –1 < x < 1 is called the interval of convergence. So we are interested in for what values this series will converge. Ex 1) For what values of x does the series converge?

Ex 2) Find a power series that converges to. For what values of x will the series converge?  a 1  r Ex 3) Find the interval of convergence for 1 + (x + 2) + (x + 2) 2 + (x + 2) 3 + … a 1 = 1 r = (x + 2)  x + 2  < 1  –1 < x + 2 < 1  –3 < x < –1 converges when

The examples so far were rational functions being represented by a power series. We can also represent other functions. Three familiar functions e x, sin x and cos x have power series that converge for all values of x. (For now, we will memorize them. They are obtained using derivatives and the rule for power series – both concepts in Calculus!)

Ex 4) Use the first seven terms of e x to evaluate e 2. Compare this to your calculator’s answer for e 2. Calculator e 2 = *pretty close within.04 If we would have used more terms, it would have been more accurate!

If we were to let the numbers in the series for e x be “pure imaginary numbers,” ix, we could combine the two and get e ix = cos x + i sin x. Since any complex number can be expressed in polar as r(cos x + i sin x) we can now say any complex number can be expressed as re ix. Ex 5) modulus: argument: polar form:

Homework #1309 Pg 733 #1–17 odd, 25–31 odd, 40, 44, 45, 46