Taylor Series and Taylor’s Theorem

1. Taylor Series and Taylor’s Theorem When is a function given by its Taylor Series?

2. So where were we? • Facts: • f is continuous and has derivatives of all orders at x = 0. • f (n)(0)=0 for all n. This tells us that the Maclaurin Series for f is zero everywhere! The Maclaurin Series for f converges everywhere, but is equal to f only at x = 0!

3. This tells us that…. Our ability write down a Taylor series for a function is not in itself a guarantee that the series will any anything to do with the function, even on its interval of convergence!

4. However, Ostebee and Zorn assures that…. “Taylor’s theorem guarantees that this unfortunate event seldom occurs.” In other words, the functions that are not given by their Taylor series are pretty weird. Most of our “everyday” functions ARE given by their Taylor Series.

5. Recall Taylor’s Theorem Suppose that f is repeatedly differentiable on an interval I containing x0 and that is the nth order Taylor polynomial based at x0. Suppose that Kn+1 is a number such that for all z in I, Then for x in I, (Page 504 in OZ)

6. Pinning this down • Recall that Pn is the nth partial sum of theTaylor Series of f based at x0. • And thus Measures the error made by Pn(x)in approximating f (x). • Taylor’s theorem gives us an upper bound on this error! The Taylor series for f will converge to f if and only if for all x | f (x) - Pn(x) | goes to zero as n →∞. Taylor’s theorem can help us establish this.

7. Using Taylor’s Theorem • Find the Taylor series for f that is based at x = p/4. • Show that this Taylor series converges to f for all values of x.

8. 1. Taylor Series for f (x) = sin(x)

9. Show that this converges to sin(x) We start with the general set-up for Taylor’s Theorem. What is Kn+1? It follows that for all x What happens to this quantity As n→∞?

10. Show that this converges to sin(x) Notice that I didn’t have to know what Pn was in order to gather this information. (In other words, our second question is independent of our first.) We start with the general set-up for Taylor’s Theorem. What is Kn+1? It follows that for all x

11. Now it’s your turn Repeat this exercise with the Maclaurin series for f (x)= cos(2x) . • Find the Maclaurin series for f (x) = cos(2x). • Show that this series converges to f for all values of x.

12. 1. Taylor Series for f (x) = cos(2x)

13. Show that this converges to cos(2x) We start with the general set-up for Taylor’s Theorem. What is Kn+1? It follows that for all x, This quantity goes to 0 as n→∞!

14. Epilogue---Two points of view Power series as functions Taylor Series First a series . . . First a function . . . . . . Then a function . . . Then a series Guarantees that f is equal to the power series where the power series converges. No a priori guarantee that f is equal to its Taylor series.

15. Why the Taylor series, then? Power series as functions Taylor Series First a series . . . First a function . . . . . . Then a function . . . Then a series If f is equal to any power series at all, that power series must be the Taylor series for f. That’s why that’s were we look! Guarantees that the power series we started with is, in fact, the TAYLOR SERIES FOR f .