Problem: Prove that this notion of convergence of f(x) dx is independent of the choice of the point a, atwhich you "cut" f(x) dx into fa f(x) dx and f f(x) dx.Guide: There are two parts you need to show, the convergent case and the divergent case. In other words, youneed to show that: Fix a real number a.(1) [Convergent case] If f f(x) dx and f f(x) dx both converge, then for all real numbers b, the improperintegrals f f(x) dr and f f(x) dx must also converge and[ f(x) dx + + √°*° f(x) dx = [*®*® f(x) dx + √°*® f(x) dx.(2) [Divergent case] If either fª f(x) dx or f f(x) dx diverge, then for all real numbers b, either f f(x) dxor f f(x) dx must also diverge.Thus after showing the two cases above, we will have proved that in the definition (the two bullet points right after(1)) of f(x) dx above, the number a can be replaced with any other real numbers.

The convergence of an improper integral depends on whether the limit defining it exists.If we're…

Answered: Problem: Prove that this notion of…

Problem: Prove that this notion of convergence of f(x) dx is independent of the choice of the point a, at which you "cut" f(x) dx into f f(x) dx and f f(x) dr. ∞

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter1: Functions

Section1.2: Functions Given By Tables

Problem 32SBE: Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it...

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Question

Problem: Prove that this notion of convergence of f(x) dx is independent of the choice of the point a, at
which you "cut" f(x) dx into fa f(x) dx and f f(x) dx.
Guide: There are two parts you need to show, the convergent case and the divergent case. In other words, you
need to show that: Fix a real number a.
(1) [Convergent case] If f f(x) dx and f f(x) dx both converge, then for all real numbers b, the improper
integrals f f(x) dr and f f(x) dx must also converge and
[ f(x) dx + + √°*° f(x) dx = [*®*® f(x) dx + √°*® f(x) dx.
(2) [Divergent case] If either fª f(x) dx or f f(x) dx diverge, then for all real numbers b, either f f(x) dx
or f f(x) dx must also diverge.
Thus after showing the two cases above, we will have proved that in the definition (the two bullet points right after
(1)) of f(x) dx above, the number a can be replaced with any other real numbers.

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