Name. Instructor K. Pernell 1. Berkeley City College Due: HW 4 - Chapter 11 - Infinite Sequences and Series. Write the first four terms of {an}.
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1 Berkeley City College Due: HW 4 - Chapter 11 - Infinite Sequences and Series Name Write the first four terms of {an}. 1) an = (-1)n n 2) an = n + 1 3n - 1 3) an = sin n! 3 Determine whether the sequence converges or diverges, and, if it converges, find lim n a n. 4) an = 9 + (0.3) n Instructor K. Pernell 1
2 5) an = 6 + 3n 1 + 7n 6) an = (-1) n 8 n 7) an = (-1) n 1-3 n 8) a + 8 n n 2
3 9) an = n! 3 n 7 n Find an explicit formula for an. 10) 6, -6, 6, -6, 6,... Indicate whether the given series converges or diverges. If it converges, find its sum. 1 11) k = 0 ( 10) k 12) k = 0 e -7k 3
4 13) k = 1 ln 10 k 14) k = 1 7 k k ) k = 0 cos k! 3 k Write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers. 16)
5 17) Solve the problem. 18) A company makes a very durable product. It sells 20,000 in the first year, but will have diminishing sales due to the product's durability, so that each year it can expect to sell only seventy-five percent of the quantity it will have sold the year before. How many of the product can the company expect to eventually sell? Use the integral test to determine the convergence or divergence of the series. 1 19) 5k- 1 k = 1 20) k = 1 cos 1/k k 2 5
6 Determine whether the sequence converges or diverges, and, if it converges, find lim n a n. 21) an = ln(4n - 4) - ln(3n - 1) Use the integral test to determine the convergence or divergence of the series. 1 22) ln 7 k = 1 k 23) k = 1 2k k Estimate the error that is made by approximating the sum of the given series by the sum of the first five terms. 1 24) k k = 1 5 6
7 Use the limit comparison test to determine if the series converges or diverges. 7 n 25) 5n 3/2 + 2n ) 3 9n - 2 ln n + 9 Use the ratio test to determine if the series converges or diverges. 2 27) n n! 28) n 9 9 n 7
8 29) 4n! n n Determine convergence or divergence of the series. (ratio test) ln n n 30) 5n + 7 Determine either absolute convergence, conditional convergence or divergence for the series. 31) (-1) n 9n + 6 ln 8n ) (-8) -n 8
9 33) (-1) n ( n n) Find the convergence set for the given power series. 34) (x - 7) n n = 0 35) n = 0 (x - 8) n 6n ) n = 0 (x - 9) n n 5 9 n 9
10 Find the radius of convergence of the power series. (x - 6) 37) n 1 + 3n n = 0 38) (x - 2) n (2n)! Find the power series representation for f(x). 39) f(x) = x 40) f(x) = e 5x 10
11 41) f(x) = e2x 1 - x Find the first few nonzero terms of the Maclaurin series for the given function. 42) f(x) = ln(1 + 2x) 43) f(x) = xe cos x Find the first three terms of the Taylor series in x - a. 44) cos x, a = 3! 2 45) e x, a = 3 11
12 46) ln x, a = 1 Find the Taylor polynomial of order 3 based at a for the given function. x 47) x + 6 ; a = -1 48) 1 x + 7 ; a = 1 49) ln(x + 1); a = 7 12
13 Answer Key Testname: MATH3B_HWCH11 1) -1, 1 2, - 1 3, 1 4 Objective: (9.1) Find Terms of Sequence 2) 1, 3 5, 1 2, ) Objective: (9.1) Find Terms of Sequence 3 2, 3 2, 0, Objective: (9.1) Find Terms of Sequence 4) 9 Objective: (9.1) Determine Convergence/Divergence and Find Limit I 5) 3 7 Objective: (9.1) Determine Convergence/Divergence and Find Limit I 6) 0 Objective: (9.1) Determine Convergence/Divergence and Find Limit I 7) Diverges Objective: (9.1) Determine Convergence/Divergence and Find Limit I 8) e 8 Objective: (9.1) Determine Convergence/Divergence and Find Limit II 9) Diverges Objective: (9.1) Determine Convergence/Divergence and Find Limit II 10) an = 6(-1) n+1 Objective: (9.1) Write Formula for nth Term of a Sequence 11) Converges; Objective: (9.2) Determine Convergence/Divergence and Find Sum 12) Converges; e 7 e 7-1 Objective: (9.2) Determine Convergence/Divergence and Find Sum 13) Diverges Objective: (9.2) Determine Convergence/Divergence and Find Sum 14) Converges; Objective: (9.2) Determine Convergence/Divergence and Find Sum 15) Converges; ) Objective: (9.2) Determine Convergence/Divergence and Find Sum Objective: (9.2) Write Repeating Decimal as Fraction 13
14 Answer Key Testname: MATH3B_HWCH11 17) Objective: (9.2) Write Repeating Decimal as Fraction 18) 80,000 Objective: (9.2) Solve Apps: Infinite Series 19) Diverges Objective: (9.3) Use Integral Test to Determine Convergence/Divergence 20) Converges Objective: (9.3) Use Integral Test to Determine Convergence/Divergence 21) ln 4 3 Objective: (9.1) Determine Convergence/Divergence and Find Limit I 22) Converges Objective: (9.3) Use Integral Test to Determine Convergence/Divergence 23) Diverges Objective: (9.3) Use Integral Test to Determine Convergence/Divergence 24) Objective: (9.3) Estimate Error in Approximating Sum 25) Diverges Objective: (9.4) Use Limit Comparison Test to Determine Convergence/Divergence 26) Diverges Objective: (9.4) Use Limit Comparison Test to Determine Convergence/Divergence 27) Converges Objective: (9.4) Use Ratio Test to Determine Convergence/Divergence 28) Converges Objective: (9.4) Use Ratio Test to Determine Convergence/Divergence 29) Converges Objective: (9.4) Use Ratio Test to Determine Convergence/Divergence 30) Converges Objective: (9.4) Determine Convergence/Divergence I 31) Diverges Objective: (9.5) Determine If Series Converges Absolutely/Conditionally or Diverges 32) Converges absolutely Objective: (9.5) Determine If Series Converges Absolutely/Conditionally or Diverges 33) Converges conditionally Objective: (9.5) Determine If Series Converges Absolutely/Conditionally or Diverges 34) 6 < x < 8 Objective: (9.6) Find Convergence Set for Power Series (Sigma Notation) 35) 7 x < 9 Objective: (9.6) Find Convergence Set for Power Series (Sigma Notation) 36) 0 x 18 Objective: (9.6) Find Convergence Set for Power Series (Sigma Notation) 37) 1 Objective: (9.6) Find Radius of Convergence of Power Series 38) Objective: (9.6) Find Radius of Convergence of Power Series 14
15 Answer Key Testname: MATH3B_HWCH11 39) x 9 + x x Objective: (9.7) Find Power Series Representation of Function 40) 1 + 5x + 25x2 2! + 125x3 3! +... Objective: (9.7) Find Power Series Representation of Function 41) 1 + 3x + 5x 2 + 3x Objective: (9.7) Find Power Series Representation of Function 42) 2x - 2x x Objective: (9.8) Find Terms in Maclaurin Series 43) ex - ex Objective: (9.8) Find Terms in Maclaurin Series 44) x - 3! 2-1 3! x - 3! ! x - 3! Objective: (9.8) Find Terms in Taylor Series 45) e (x - 3) (x - 3) Objective: (9.8) Find Terms in Taylor Series 46) (x - 1) (x - 1) (x - 1) Objective: (9.8) Find Terms in Taylor Series 47) (x + 1) (x + 1)2 + Objective: (9.9) Find Taylor Polynomial 48) x (x - 1) (x - 1) Objective: (9.9) Find Taylor Polynomial 49) ln 8 + x (x - 7) (x - 7) Objective: (9.9) Find Taylor Polynomial 6 (x + 1)
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