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4. The function f : [1, ∞) → R defined by f(x) = (1-x2) ez is continuous and decreasing on its domain. Define the series S and the integral I by s=Σ f(n) n=1 and I = [ f(x) dx. (a) Explain why the integral I is improper. (b) Using appropriate mathematical rigour, show that that the integral I diverges. (c) Explain why the integral test is not applicable in this case for determining whether the series S converges or diverges.

4. The function f : [1, ∞) → R defined by f(x) = (1-x2) ez is continuous and decreasing on its domain. Define the series S and the integral I by s=Σ f(n) n=1 and I = [ f(x) dx. (a) Explain why the integral I is improper. (b) Using appropriate mathematical rigour, show that that the integral I diverges. (c) Explain why the integral test is not applicable in this case for determining whether the series S converges or diverges.

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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4. The function f : [1, ∞) → R defined by f(x) = (1-x2)ex is continuous and decreasing on its domain. Define the series S and the integral I by ∞ S = - Σf(n) n=1 and I = f(x) dx. (a) Explain why the integral I is improper. (b) Using appropriate mathematical rigour, show that that the integral I diverges. (c) Explain why the integral test is not applicable in this case for determining whether the series S converges or diverges. (d) Determine whether the series S converges or diverges.
Transcribed Image Text:4. The function f : [1, ∞) → R defined by f(x) = (1-x2)ex is continuous and decreasing on its domain. Define the series S and the integral I by ∞ S = - Σf(n) n=1 and I = f(x) dx. (a) Explain why the integral I is improper. (b) Using appropriate mathematical rigour, show that that the integral I diverges. (c) Explain why the integral test is not applicable in this case for determining whether the series S converges or diverges. (d) Determine whether the series S converges or diverges.
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