The series converges by the Alternating Series Test. Use Theorem 9.9: Error Bounds for Alternating Series to find how many terms give a partial sum, Sn, within 0.01 of the sum, S, of the series.
Theorem 9.9: Error Bounds for Alternating Series Let n = Σ Suppose that 0
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The series converges by the Alternating Series Test. Use Theorem 9.9: Error Bounds for Alternatin...
1. (Alternating Series Test.) This shows that for this particular sort of alternating series, the error...
1. (Alternating Series Test.) This shows that for this particular sort of alternating series, the error in approximating the infinite sum by a partial sum is at most the first omitted term. Suppose that aj > a2 > a3 > ... > 0 and that limnyoo An = 0. Let sn = {k=1(-1)kak. (a) Prove that if n > m > 0 then |sn – Sm! < am+1. (b) Prove that 2-1(-1)kak converges and that, for all n > 0,...
Study: Ch. 5 5.2 #93-96, 5.5 280-285 The given series converges by Alternating Series Test. Use...
Study: Ch. 5 5.2 #93-96, 5.5 280-285 The given series converges by Alternating Series Test. Use the estimate |RN| <bn+1 to find the least value of N that guarantees that the sum Sy differs from the infinite sum n n=1 by at most an error of 0.01. Answer (a) What is N? (b) What is Sy and what is the actual sum S of the series? (c) Is S - SN <0.01?
I need help on proving problem b. (-1)"+1a, be an alternat- Theorem 7–16 (Alternating Series Test):...
I need help on proving problem b.
(-1)"+1a, be an alternat- Theorem 7–16 (Alternating Series Test): Let ing series such that (i) anan+1 > O for every n. (ii) lima, = 0. Then (-1)*+la, and (-1)"a, converge. (b) Let [(-1)"+1 an be an alternating series satisfying the hypotheses of Theorem 7–16 and converging to L. If {Sn} is the sequence of partial sums associated with the series, then |L-S <|an+1l.
Find a formula for the nth partial sum of the series and use it to determine...
Find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If the series converges, find its sum. 10 Σ 10 n+1 n n=1 Sn If the series converges, what is its sum? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The series sum is (Type an integer or a fraction.) B. The series diverges.
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...
6. For each given series, complete the following tasks: (i) Prove that the series converges ab- s...
6. For each given series, complete the following tasks: (i) Prove that the series converges ab- solutely; (i) Show that the series satisfies all conditions of the Alternating Series Test; (ii) Find the partial sum sy of the series, and then estimate its remainder Ra: (iv) Determine how many terms are needed to approximate the sum of the series accurate to within 0.001, and then find this approximation. (a) L (b) Σ 27! 6. For each given series, complete the...
Use the sum of the first ten terms to approximate the sum of the series -Estimate the error by ta...
use the sum of the first ten terms to approximate the sum of the series -Estimate the error by takingthe average of the upper (Hint: Use trigonometric substitution, Round your answers to three decimal places Theorem 16. Remainder Estimate for the Integral Test Let f(x) be a positive-valued continuous decreasing function on the interval [I,0o) such that f(n): an for every natural number n. lf the series Σ an converges, then f(x)dx s R f(x)dx
use the sum of the...
E) The series Σ-(-1)" 2- n a. converges conditionally. b. diverges by the nth term test. c. converges absolutely, d. converges by limit comparison test. F) The sum of the series 2-3)" is equa...
E) The series Σ-(-1)" 2- n a. converges conditionally. b. diverges by the nth term test. c. converges absolutely, d. converges by limit comparison test. F) The sum of the series 2-3)" is equal to e. None of the above
E) The series Σ-(-1)" 2- n a. converges conditionally. b. diverges by the nth term test. c. converges absolutely, d. converges by limit comparison test. F) The sum of the series 2-3)" is equal to e. None of the above
how to find the actual sum and how to find the maxinmum error, do we have any formula? thanks 11 Let *(3n+1) Suppose we estimated Σ a" by computing the partial sum k-1-2+. According to the Altern...
how to find the actual sum and how to find the maxinmum error,
do we have any formula? thanks
11 Let *(3n+1) Suppose we estimated Σ a" by computing the partial sum k-1-2+. According to the Alternating Series Estimation Theorem, (ak is an undenestimate, and the maximumerror is 12 (b) is an overestimate, and the maximum error is 24 (e) k is an overestimate, and the maximum error is 12 (d) The Alternating Series Estimation Theorem cannot be used because...
6. We want to use the Integral Test to show that the positive series a converges....
6. We want to use the Integral Test to show that the positive series a converges. All of the following need to be done except one. Which is the one we don't need to do? (a) Find a function f(x) defined on [1,00) such that f(x) > 0, f(x) is decreasing, and f(n) = a, for all n. (b) Show that ſ f(z) dr converges. (e) Show that lim Ss6 f(x) dx exists. (d) Show that lim sexists. 7. Suppose...
1. (Alternating Series Test.) This shows that for this particular sort of alternating series, the error in approximating the infinite sum by a partial sum is at most the first omitted term. Suppose that aj > a2 > a3 > ... > 0 and that limnyoo An = 0. Let sn = {k=1(-1)kak. (a) Prove that if n > m > 0 then |sn – Sm! < am+1. (b) Prove that 2-1(-1)kak converges and that, for all n > 0,...
Study: Ch. 5 5.2 #93-96, 5.5 280-285 The given series converges by Alternating Series Test. Use the estimate |RN| <bn+1 to find the least value of N that guarantees that the sum Sy differs from the infinite sum n n=1 by at most an error of 0.01. Answer (a) What is N? (b) What is Sy and what is the actual sum S of the series? (c) Is S - SN <0.01?
I need help on proving problem b.
(-1)"+1a, be an alternat- Theorem 7–16 (Alternating Series Test): Let ing series such that (i) anan+1 > O for every n. (ii) lima, = 0. Then (-1)*+la, and (-1)"a, converge. (b) Let [(-1)"+1 an be an alternating series satisfying the hypotheses of Theorem 7–16 and converging to L. If {Sn} is the sequence of partial sums associated with the series, then |L-S <|an+1l.
Find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If the series converges, find its sum. 10 Σ 10 n+1 n n=1 Sn If the series converges, what is its sum? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The series sum is (Type an integer or a fraction.) B. The series diverges.
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...
6. For each given series, complete the following tasks: (i) Prove that the series converges ab- solutely; (i) Show that the series satisfies all conditions of the Alternating Series Test; (ii) Find the partial sum sy of the series, and then estimate its remainder Ra: (iv) Determine how many terms are needed to approximate the sum of the series accurate to within 0.001, and then find this approximation. (a) L (b) Σ 27! 6. For each given series, complete the...
use the sum of the first ten terms to approximate the sum of the series -Estimate the error by takingthe average of the upper (Hint: Use trigonometric substitution, Round your answers to three decimal places Theorem 16. Remainder Estimate for the Integral Test Let f(x) be a positive-valued continuous decreasing function on the interval [I,0o) such that f(n): an for every natural number n. lf the series Σ an converges, then f(x)dx s R f(x)dx
use the sum of the...
E) The series Σ-(-1)" 2- n a. converges conditionally. b. diverges by the nth term test. c. converges absolutely, d. converges by limit comparison test. F) The sum of the series 2-3)" is equal to e. None of the above
E) The series Σ-(-1)" 2- n a. converges conditionally. b. diverges by the nth term test. c. converges absolutely, d. converges by limit comparison test. F) The sum of the series 2-3)" is equal to e. None of the above
how to find the actual sum and how to find the maxinmum error,
do we have any formula? thanks
11 Let *(3n+1) Suppose we estimated Σ a" by computing the partial sum k-1-2+. According to the Alternating Series Estimation Theorem, (ak is an undenestimate, and the maximumerror is 12 (b) is an overestimate, and the maximum error is 24 (e) k is an overestimate, and the maximum error is 12 (d) The Alternating Series Estimation Theorem cannot be used because...
6. We want to use the Integral Test to show that the positive series a converges. All of the following need to be done except one. Which is the one we don't need to do? (a) Find a function f(x) defined on [1,00) such that f(x) > 0, f(x) is decreasing, and f(n) = a, for all n. (b) Show that ſ f(z) dr converges. (e) Show that lim Ss6 f(x) dx exists. (d) Show that lim sexists. 7. Suppose...
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