To get the same accuracy on a larger interval would

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Essential Calculus: Early Transcendentals
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified
To get the same accuracy on a larger interval would require more terms. Exercises 11.11. 1. Find a polynomial approximation for cos x on [0 , π ], accurate to ± 10 3 2. How many terms of the series for ln x centered at 1 are required so that the guaranteed error on [1 / 2 , 3 / 2] is at most 10 3 ? What if the interval is instead [1 , 3 / 2]? 3. Find the first three nonzero terms in the Taylor series for tan x on [ π / 4 , π / 4], and compute the guaranteed error term as given by Taylor’s theorem. (You may want to use Sage or a similar aid.)
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Essential Calculus: Early Transcendentals
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified
294 Chapter 11 Sequences and Series 4. Show that cos x is equal to its Taylor series for all x by showing that the limit of the error term is zero as N approaches infinity. 5. Show that e x is equal to its Taylor series for all x by showing that the limit of the error term is zero as N approaches infinity. These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way. Determine whether the series converges. 1. n =0 n n 2 + 4 2. 1 1 · 2 + 1 3 · 4 + 1 5 · 6 + 1 7 · 8 + · · · 3. n =0 n ( n 2 + 4) 2 4. n =0 n ! 8 n 5. 1 3 4 + 5 8 7 12 + 9 16 + · · · 6. n =0 1 n 2 + 4 7. n =0 sin 3 ( n ) n 2 8. n =0 n e n 9. n =0 n ! 1 · 3 · 5 · · · (2 n 1) 10. n =1 1 n n 11. 1 2 · 3 · 4 + 2 3 · 4 · 5 + 3 4 · 5 · 6 + 4 5 · 6 · 7 + · · · 12. n =1 1 · 3 · 5 · · · (2 n 1) (2 n )! 13. n =0 6 n n ! 14. n =1 ( 1) n 1 n
11.12 Additional exercises 295 15. n =1 2 n 3 n 1 n ! 16. 1 + 5 2 2 2 + 5 4 (2 · 4) 2 + 5 6 (2 · 4 · 6) 2 + 5 8 (2 · 4 · 6 · 8) 2 + · · · 17. n =1 sin(1 /n ) Find the interval and radius of convergence; you need not check the endpoints of the intervals. 18. n =0 2 n n ! x n 19. n =0 x n 1 + 3 n 20. n =1 x n n 3 n 21. x + 1 2 x 3 3 + 1 · 3 2 · 4 x 5 5 + 1 · 3 · 5 2 · 4 · 6 x 7 7 + · · · 22. n =1 n ! n 2 x n 23. n =1 ( 1) n n 2 3 n x 2 n 24. n =0 ( x 1) n n ! Find a series for each function, using the formula for Maclaurin series and algebraic manip- ulation as appropriate. 25. 2 x 26. ln(1 + x ) 27. ln 1 + x 1 x 28. 1 + x 29. 1 1 + x 2 30. arctan( x ) 31. Use the answer to the previous problem to discover a series for a well-known mathematical constant.

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