1. The Convergence Problem Recall that the nth Taylor polynomial for a function f about x = x o has the property that its value and the values of its first n derivatives match those of f at x o. As n increases, more and more derivatives match up, so it is reasonable to hope that for values of x near x o the values of the Taylor polynomials might converge to the value of f(x); that is

2. The Convergence Problem However, the nth Taylor polynomial for f is the nth partial sum of the Taylor series for f, so the formula below is equivalent to stating that the Taylor series for f converges at x, and its sum is f(x).

3. The Convergence Problem

4. One way to show that this is true is to show that However, the difference appearing on the left side of this equation is the nth remainder for the Taylor series. Thus, we have the following result.

5. The Convergence Problem Theorem:

6. The Remainder Estimation Theorem If the function f can be differentiated n+1 times on an interval I containing the number x o, and if M is an upper bound for then for all in x.

7. Example Show that the Maclaurin series for cosx converges to cosx for all x; that is From Theorem 10.9.2, we must show that for all x as. For this purpose let f(x) = cosx, so that for all x, we have or In all cases, we have so we will say that M = 1 and x o = 0 to conclude that

8. However, it follows that So becomes

9. Example Use the Maclaurin series for sinx to approximate sin 3 o to five decimal-point accuracy. In the Maclaurin series The angle is assumed to be in radians. Since 3 o =  /60 it follows that

10. In the Maclaurin series We must now determine how many terms in the series are required to achieve five decimal-place accuracy. We have two choices. 1.The remainder estimation theorem For five decimal-place accuracy, we need

11. Let f(x)= sinx, then f(x) is either or and in either case Thus with M = 1, x =  /60, and x o = 0, we have

18. Example Show that the Maclaurin series for e x converges to e x for all x; that is

28. Binomial Series If m is a real number, then the Maclaurin series for (1 + x) m is called the binomial series; it is given by In the case where m is a nonnegative integer, the function f(x) = (1 + x) m is a polynomial of degree m, so The binomial series reduces to the familiar binomial expansion

29. Binomial Series It can be proved that if m is not a nonnegative integer, then the binomial series converges to (1 + x) m if |x| < 1. Thus, for such values of x or in sigma notation

30. Example Find the binomial series for (a) (b)