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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following. Σ²²³(²-)² n=1 (a) Use the Ratio Test to verify that the series converges n a lim n→∞ an Sn n+ 1 n = n (b) Use a graphing utility to graph the first 10 terms of the sequence of partial sums. Use a graphing utility to find the indicated partial sum S, and complete the table. (Round your answer to four decimal places.) < 1 10 15 20 25 (c) Use the table to estimate the sum of the series. (Round your answer to the nearest integer.) (d) Explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series. The more slowly the terms of the series approach 0, the more rapidly the sequence of the partial sums approaches the sum of the series. The more rapidly the terms of the series approach the sum of the series, the more slowly the sequence of the partial sums approaches 0. The more rapidly the terms of the series approach 0, the more slowly the sequence of the partial sums approaches the sum of the series. The more rapidly the terms of the series approach the sum of the series, the more rapidly the sequence of the partial sums approaches 0. The more rapidly the terms of the series approach 0, the more rapidly the sequence of the partial sums approaches the sum of the series.
Transcribed Image Text:Consider the following. Σ²²³(²-)² n=1 (a) Use the Ratio Test to verify that the series converges n a lim n→∞ an Sn n+ 1 n = n (b) Use a graphing utility to graph the first 10 terms of the sequence of partial sums. Use a graphing utility to find the indicated partial sum S, and complete the table. (Round your answer to four decimal places.) < 1 10 15 20 25 (c) Use the table to estimate the sum of the series. (Round your answer to the nearest integer.) (d) Explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series. The more slowly the terms of the series approach 0, the more rapidly the sequence of the partial sums approaches the sum of the series. The more rapidly the terms of the series approach the sum of the series, the more slowly the sequence of the partial sums approaches 0. The more rapidly the terms of the series approach 0, the more slowly the sequence of the partial sums approaches the sum of the series. The more rapidly the terms of the series approach the sum of the series, the more rapidly the sequence of the partial sums approaches 0. The more rapidly the terms of the series approach 0, the more rapidly the sequence of the partial sums approaches the sum of the series.
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