Comparing lntegrals Let f be continuous on the interval [a, ∞) . Show that if the i

Comparing lntegrals Let f be continuous on the...

Key Concepts

Absolute Convergence

Absolute convergence refers to the situation where the integral of the absolute value of a function converges. This concept is crucial because if an integral converges absolutely, then it guarantees the convergence of the original integral, even if the function takes both positive and negative values.

Comparison Test

The comparison test is a method used to determine the convergence of an integral by comparing it to another integral with a known convergence behavior. If the absolute value of the integrand is bounded by a function whose integral converges, then the integral of the original function will also converge.

Improper Integrals

An improper integral is defined as an integral where either the interval of integration is unbounded (extending to infinity) or the integrand has an infinite discontinuity within the interval. In these cases, the integral is interpreted as a limit, and convergence depends on whether this limit exists and is finite.

Transcript

00:01 This question asks us to prove an interesting variation on the comparison test we've seen for integrals.

00:08 So recall that if 0 is less than g of x, so g of x is a positive function, and f of x is a bigger function, and if, moreover, the integral from some starting point to infinity, or f of x, the x converges, then also the integral for g of x converges.

00:40 This would call the comparison test.

00:43 Now, note that this only works for positive functions.

00:47 Otherwise, you could just do something with a function that goes to negative infinity, and the rule breaks.

00:54 In this case, we're asked to prove a variation on this.

01:02 If the integral from a to infinity for the absolute value of fx, tx, converges, then the integral for the original function converges as well.

01:24 Note that this doesn't hold the other way around.

01:26 If fx converges as an integral, the absolute value might not, because you might have things under the x -axis that cancel out things above the x -axis.

01:38 But this says if the absolute value converges, then the thing itself converges.

01:44 So we're going to prove this using our previous version of the computer comparison test.

01:51 And we're going to look at the function fx plus the absolute value of fx.

02:04 Now certainly this is positive because, well, basically we consider two cases.

02:11 If f itself is already positive, then we have something positive plus something positive, definitely positive.

02:21 If f is negative, however, f of x is is exactly the same as f only positive.

02:29 So if you add negative something plus that thing, you get exactly zero.

02:35 So it's never less than zero, which is the important part.

02:39 And on the other hand, this guy is less than two times the absolute value of f x.

02:47 Again, we can consider the sort of separate cases.

02:51 If f is positive, then this is exactly quality.

02:54 If f is negative, then this is a strict inequality, because something negative plus something positive is definitely less than twice a positive thing.

03:09 So we can do the classic convergence test using these guys.

03:17 So we say, well, we know a very simple and straightforward fact is that because a to infinity, f of x, the s, x is convergent.

03:35 This is not the comparison test yet.

03:37 This is just sort of basic integration.

03:47 The integral for twice ff x also converges, and it equals twice the integral of this guy.

04:02 This is a basic fact.

04:03 That's very easy to prove.

04:06 Ok, so now we are in the situation of our original comparison test.

04:12 We can show that, so why the comparison test, the integral of a to infinity f of x plus the absolute value of f x, dx converges.

04:31 So now we're gonna do the sort of major trick, which uses something else that we haven't technically proven.

04:41 If the integral from, let's say, a, to infinity of some function, let's call it h of x, dx...

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