378 Chapter 10 Infinite Series e = 1 + x + + (for all x) + 4! (10.14) 2! 3! کہ sin xX- (for all x) (10.15) 3! 5! (for all x) (10.16) 4! COS X =- 2! e In (1 + x) = x- 2 + (-1<x<I) (10.17) 3 4 Some additional series will be obtained in the next section. Exercises / Section 10.3 2 8. In(1-x) --- 2 3 In Exercises 1-11, verify the Maclaurin series expansions. Con- firm the results using your graphing utility. r 1. sinx=r 3! 5! 9, sinh ra -e)=x+ + bile . cosa x H +)-++... 2. cos x = 1 - + 2! 4! 10. = 1 + 2! 4! 3' 3. sin 3r3r- + 11. = 1 + x + + + ... 1-X x1 <1 31" 4 4. cos 4xl- 21+ 12. (Optional) Show that the Series (10.15) and (10.16) converge for all 13. Verify the series expansion 2! 3! 4! (1 - x)2 = (n + 1)x" 5. c** - 1 -*+ +++... (1 = +... 6.2 = 1 + 2x + + 20 + 47* 31" by (a) using the binomial theorem; (b) finding the Maclaurin series expansion; (e) dividing out 1/(1-x)? 7. In(1 + x)x- + 2 3 10.4 Operations with Series In this section we will study a number of operations that yield new series from series already known. Example 1 Find the Maclaurin series for sin r. Solution. Consider the series x sin .xx- 3! 5! +
10.4 Operations with Series 379 from the last section. If we replace x by r°, we obtain sin rer? (x)} (x?) 3! 5! + or 10 + sin = 3! 5! Since this series is a power series, it must be the Maclaurin series of sin x, since such expansions are unique. Example 2 Find the Maclaurin series for xe'. Solution. From + = 1 + x + 2! 3! we obtain by direct multiplication xel- (1 1 + x + + +) + 2! 3! =Y+ 2! + + 3! It has already been noted that convergent power series can be differentiated termwise; the same is true of integration. Example 3 Show that (d/dx)e' = e' by the use of Maclaurin series. Solution. d dx 2 1 + x + + + 2! 3! 4! ...) 2x 3r = 0 + 1 + + + 2! 3! 4! = 1 + x + + 2! 3! Example Find the Maclaurin series of Arcsin x by integrating (d/dx) Arcsin x termwise. Solution. We recall that d 1 Arcsin x - dx VI-
b) Using the given Maclaurin series on page 378, section 10.3 of your textbook determine the first 5 terms of бе2x = y = , in expanded form, and simplify your result. X2
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