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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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