Clicker Question 1 The series A. converges absolutely. B. converges conditionally but not absolutely. C. converges absolutely but not conditionally. D. diverges.

Clicker Question 2 The series A. converges absolutely. B. converges conditionally but not absolutely. C. converges absolutely but not conditionally. D. diverges.

Power Series (4/16/14) Series need not be built out of numbers only. They can have variables in them, and therefore be functions. So, what about the very simple series: 1 + x + x2 + x3 + x4 + …. For what x’s does it converge?? To what? Such a series, with a variable raised to successive powers, is called a power series. These are polynomials of “infinite degree”!

More on this simple example So 1 + x + x2 + x3 + x4 + …. converges as long as |x| < 1. We say then that the radius of convergence of this series is 1. We also say that the interval of convergence of this series is the open interval -1 < x < 1 Note that this series does not converge at the endpoints of this interval (when x = 1).

Another example Consider the slightly more complicated power series x / 1 + x2 / 2 + x3 / 3 + x4 / 4 + …. For what x’s does this one converge? Since this one’s not geometric, the answer is not obvious. Try some values of x… For example, does it converge at x = 0? 1? -1? Perhaps we can use the ratio test? Try it!

…and another example How about the series 1 + x / 1! + x2 / 2! + x3 / 3! + x4 / 4! + …. ? What does it converge to at x = 0? 1? -1? Use the ratio test to determine its radius of convergence and interval of convergence. Want to guess the value of this infinite series for all x?

General Definition of Power Series The examples we have looked at so far have been power series “centered at 0” (i.e., the interval of convergence has its center at 0). In general a power series has the form c0 + c1(x – a) + c2(x – a)2 + c3(x – a)3 + …. Such a series is “centered at a”. Example: Find the interval of convergence of (x – 3) + 2(x – 3)2 + 3(x – 3)3 + ….

Assignment for Friday Read Section 11.8. On page 745, do Exercises 1, 2, 3 – 19 odd. Work on Hand-in #4, which is due on Monday at class time.