pow_series_for_funct

# pow_series_for_funct - (12.9 Power Series for Functions...

• Notes
• 4

This preview shows pages 1–2. Sign up to view the full content.

(12.9) Power Series for Functions Recall the formula for geometric series: n =0 ar n = a 1 - r provided | r | < 1. (We really mean, n =0 ar n converges to a 1 - r provided | r | < 1.) So if we let a = 1 and r = x , we have n =0 x n = 1 + x + x 2 + x 3 + · · · = 1 1 - x provided | x | < 1, or - 1 < x < 1. So we think of f ( x ) = n =0 x n as a function of x . We therefore know is that it is really the function f ( x ) = 1 1 - x , but only for - 1 < x < 1 (for x outside that interval, the function f ( x ) is undefined). We can use this series to find power series for other functions. Example Find a power series for the function f ( x ) = 2 x x - 3 . We start with 1 1 - x = 1 + x + x 2 + x 3 + · · · . Trick: substitute x 3 for x . We have 1 1 - x 3 = 1 + x 3 + x 3 2 + x 3 3 + · · · or 3 3 - x = 1+ x 3 + x 2 9 + x 3 27 + · · · . We adjust this a little (by dividing by 3 then multiplying by - 1): 1 x - 3 = - 1 3 - x 9 - x 2 27 - x 3 81 - · · · . This is 1 x - 3 = n =0 - x n 3 n +1 . Now multiply by 2 x to get the desired result: 2 x x - 3 = - 2 x 3 - 2 x 2 9 - 2 x 3 27 - 2 x 4 81 - · · · = n =0 - 2 x n +1 3 n +1 = n =0 - 2 x n 3 n . So we are done, but we should determine the interval of convergence. The geometric series 3 3 - x = 1 + x 3 + x 2 9 + x 3 27 + · · · converges if x 3 < 1 or - 3 < x < 3. Multiplying by 2 x doesn’t affect convergence. So we conclude that

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern