Let f be a positive, continuous, and decreasing function for x ≥ 1, such that an = f(n). If the series ∞ an n = 1 converges to S, then the remainder RN = S − SN is bounded by 0 ≤ RN ≤ ∞ N f(x) dx. Use the result above to approximate the sum of the convergent series using the indicated number of terms. (Round your answers to four decimal places.) ∞ n = 1 1 n2 + 1 , twelve terms 0.9969 Correct: Your answer is correct. Include an estimate of the maximum error for your approximation. 0.06379 Incorrect: Your answer is incorrect. Need Help? Read It
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Answer: 0.9969
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Answer: 0.0831
Let f be a positive, continuous, and decreasing function for x ≥ 1, such that an...
Let Pbé à pósitive, continuous, and decreasing function for x 2 1, such that an-n). If the series an n 1 converges to S, then the remainder RN -S-Sw is bounded by Use the result above to approximate the sum of the convergent series using the indicated number of terms. (Round your answers to four decimal place Σ ,T, ten terms n2 +1' Include an estimate of the maximum error for your approximation. (Give your answer to four decimal places.)...
Let f be a positive, continuous, and decreasing function for x 2 1, such that a, = f(n). Note that if the series, converges to S, then the remainder R - S - Sis bounded by OSRNS / (x) dx. Use these results to find the smallest N such that RN 30.001 for the convergent series.
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...
3. Suppose lim s(a) dr = co, where f(a) is a positive, decreasing and continuous function. Which of the following statements is true about the series f(n)? Choose one. n=1 *Please write the letter of your choice. (a) The series converges too. (b) The series converges, but not necessarily to o. (c) The series diverges. (d) The given information is not enough to determine if the series converges or diverges.
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(1 point) Consider the following convergent series: Suppose that you want to approximate the value of this series by computing a partial sum, then bounding the error using the integral remainder estimate. In order to bound the value of the series between two numbers which are no more than 10 apart, what is the fewest number of terms of the series you would need? Fewest number of terms is 585 (1 point) Consider the following series: le(n Use...
QUESTION 5 2x Given that f(x)= is continuous and decreasing on [3,+). x2 +4 Determine the convergence of x2 2x i) dx. 3 +00 2n ii) State the convergence of the series - Justify your answer. n=3 n° +4
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Consider the following alternating series. (-1)*+ 1 3k k=1 (a) Show that the series satisfies the conditions of the Alternating Series Test. 1 3" Since lim o and an + 1 for all n, the series is convergent (b) How many terms must be added so the error in using the sum S, of the first n terms as an approximation to the sum n=10 X (c) Approximate the sum of the series so that the error is less than...
1. Answer the following questions. Justify your answers. a. (8pts) Find the Taylor series for f(x) = (5x centered at a = 1 using the definition of the Taylor series. Also find the radius of convergence of the series. b. (8pts) Find a power series representation for the function f(x) = 1 5+X C. (4pts) Suppose that the function F is an antiderivative of a function f. How can you obtain the Maclaurin series of F from the Maclaurin series...