Calculus II Chapter 9 Review Name______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. an 1) a = 1, a = 1) 1 n+1 n + 2

Find a formula for the nth term of the sequence. 1 1 1 1 2) 1, - , , - , 2) 4 9 16 25

Find the limit of the sequence if it converges; otherwise indicate divergence. 9 + 8n 3) a = 3) n 9 + 5n

3 4) (-1)n 1 - 4) n

Determine if the sequence is bounded. Δn 5) 5) 4n

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 3 3 3 3 6) 3 - + - + ... + (-1)n-1 + ... 6) 8 64 512 8n-1

7 7 7 7 7) + + + ... + + ... 7) 1·3 2·4 3·5 n(n + 2)

Find the sum of the series. Q 9 8) _ (-1)n 8) 4n n=0

Use partial fractions to find the sum of the series. 3 9) 9) _ (4n - 1)(4n + 3) 1

Determine if the series converges or diverges; if the series converges, find its sum. 3n+1 10) _ 10) 7n-1

1 Q cos nΔ 11) _ 11) 7n n=0

Find the values of x for which the geometric series converges.

12) _ -3n xn 12)

Find the sum of the geometric series for those x for which the series converges.

13) (-1)n (6x)2n 13)

14) _ (x + 2)n 14)

Use the integral test to determine whether the series converges. 3 15) 15) _ n 1

7 16) _ 16) ex - 1

Use the direct comparison test to determine if the series converges or diverges. Q 10 17) 17) _ 3 n=1 7 n + 8 n

5 + 2 cos n 18) _ 18) n4

Use the limit comparison test to determine if the series converges or diverges. 5 + 8 sin n 19) _ 19) 9n5/4 + 5 cos n

1 20) 20) _ 4 + 7n ln n

Use the ratio test to determine if the series converges or diverges.

21) n! e-6n 21)

2 Q 9(n!)2 22) 22) _ (2n)! n=1

Use the root test to determine if the series converges or diverges. ln n n 23) 23) _ 3n - 10

n 24) _ 24) 5n1/n - 1 n

Determine convergence or divergence of the series. 6 25) _ 25) (5n + 8)4/3

3n1/2 26) _ 26) 4n3/2 + 2

Determine if the series defined by the formula converges or diverges. 8 + sin n 27) a = 2, a = a 27) 1 n+1 n n

n 28) a 28) n + 1 n

Determine convergence or divergence of the alternating series. (-1)n 29) _ 29) 6 4 + 2

5n6 + 6 30) _ (-1)n 30) 3n2 + 6

Determine either absolute convergence, conditional convergence or divergence for the series. 2 5 31) 31) 6 6 8

5n + 7 32) ln 32) _ 5n + 6

3 Q (n!)27n 33) (-1)n 33) _ (5n + 1)! n=1

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. 1 34) _ +1 34) 4n

Find the infinite sum accurate to 3 decimal places. +Q 1 35) (-1)n+1 35) _ (2n)! = 1

Find the interval of convergence of the series. (x - 4)n 36) 36) _ 6n + 8 0

7 2n 37) _ 37) 4n

For what values of x does the series converge absolutely?

38) _ (8x + 3)n 38)

For what values of x does the series converge conditionally? 2n xn 39) 39) _ n!

40) 40) _ n

Find the sum of the series as a function of x.

41) _ (x - 6)n 41)

x 3 n 42) 42) _ 6

Find the Taylor polynomial of order 3 generated by f at a. 1 43) f(x) = , a = 0 43) x + 2

4 44) f(x) = ln(x + 1), a = 7 44)

Find the Maclaurin series for the given function. 45) e6x 45)

1 46) 46) x - 2

47) sin 7x 47)

Find the Taylor series generated by f at x = a. 1 48) f(x) = , a = 4 48) x

49) f(x) = e4x, a = 8 49)

Find the quadratic approximation of f at x = 0. 50) f(x) = 64 - x2 50)

Use substitution to find the Taylor series at x = 0 of the given function. 51) -4x 51)

Find the Taylor series for the given function. 52) x7 ex 52)

53) cos2 2x 53)

1 54) f(x) = 54) (1 - 7x)2

Solve the problem. 55) The polynomial 1 + 7x + 21x2 is used to approximate f(x) = (1 + x)7 on the interval 55) -0.01 K x K 0.01. Use the Remainder Estimation Theorem to estimate the maximum absolute error.

Find the Taylor polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3. x sin t 56) F(x) = dt , [0, 1] 56) a t 0

Use series to evaluate the limit. e-5x - 1 57) lim 57) x0 x

5 58) lim x2 e2/x2 - 1 58) xQ

Find the first four terms of the binomial series for the given function. 59) (1 + 6x)1/2 59)

x -2/3 60) 1 - 60) 3

Solve the problem. 1 d 1 61) Use the binomial series and the fact that tanh-1 x = to generate the first 61) dx a 1- x2 -1 four nonzero terms of the Taylor series for tanh-1 x. Use the first three terms of the series to estimate tanh-1 (0.5).

6 Answer Key Testname: CYCALCH.9R

1 1 1 1 1) 1, , , , 3 12 60 360 (-1)n+1 2) an = n2 8 3) 5 4) Diverges 5) bounded 1 31 - n (-8) 8 6) ; 1 3 1 + 8 7n(3n + 5) 21 7) ; 4(n + 1)(n + 2) 4 36 8) 5 1 9) 4 63 10) Converges; 4 7 11) Converges; 8 1 12) |x| < 3 1 13) 1 + 36x2 1 14) -1 - x 15) diverges 16) converges 17) Diverges 18) Converges 19) Converges 20) Diverges 21) Diverges 22) Converges 23) Converges 24) Converges 25) Converges 26) Diverges 27) Converges 28) Diverges 29) Converges 30) Diverges 31) Converges absolutely 32) Converges conditionally 7 Answer Key Testname: CYCALCH.9R

33) diverges 34) 9.77 × 10-4 35) 0.460 36) 3 ≤ x < 5 37) 5 < x < 9 1 1 38) - < x < - 2 4 39) None 40) x = 3 1 41) - x - 7 6 42) - x - 9 1 x x2 x3 43) - + - 2 4 8 16 x - 7 (x - 7)2 (x - 7)3 44) ln 8 + - + 8 128 1536 ∞ 6n xn 45) ∑ n! n=0 ∞ xn 46) - ∑ 2n+1 n=0 ∞ (-1)n 72n+1 x2n+1 47) ∑ (2n + 1)! n=0 ∞ (-1)n (x - 4)n 48) ∑ 4n+1 n=0 ∞ e32 4n (x - 8)n 49) ∑ n! n=0 x2 50) Q(x) = 8 - 16 ∞ (-1)n(4)nxn 51) ∑ n! n=0 ∞ xn+7 52) ∑ n! n=0 ∞ 1 (-1)n (4)2n x2n 53) + 2 ∑ 2(2n)! n=0 ∞ 54) ∑ (n+1) 7n xn n=0 55) ≈ 3.642 x 10-5

8 Answer Key Testname: CYCALCH.9R

x3 x5 56) x - + 18 600 57) -5 58) 2 9 27 59) 1 + 3x - x2 + x3 2 2 2 5 40 60) 1 + x + x2 + x3 9 81 2187 x3 x5 x7 61) x + - + ; 0.535 3 5 7