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While going through an Indian text on Analysis I found a test for convergence of improper integral. It was stated without proof. I tried to prove it...then some doubts pop up...

Statement is this:

Let $f(x)$ be bounded and integrable in every closed subinterval of $ (a,\infty)$, where $a >0$. Let $\mu$ be a positive number such that $\lim_{x \rightarrow \infty} x^{\mu}f(x)$ exists. If $\mu > 1 $, then $\int_{a}^{\infty} f(x)dx$ converges. If $ \mu \leq 1 $, then $\int_{a}^{\infty} f(x)dx$ diverges.

My proof:

$\lim_{x \rightarrow \infty} x^{\mu}f(x) =L $ then for suitable $\epsilon >0$ we will get $x_{0} $ such that $(-L+\epsilon)< |x|^{\mu}|f(x)| < (L- \epsilon)$ for $x>x_{0}$.

This will lead to to $|f(x)| < (L- \epsilon)|x|^{-\mu}$ then using comparison test we will get as $\mu$>1 $\int_{a}^{\infty} f(x)dx$ converges absolutely.

But this proof cannot be used for discussing divergence. Even though we use left inequality $|x|^{-\mu}(-L+\epsilon)< |f(x)|$ .

There are two problems I felt. One is comparison test is applicable for positive functions. $|x|^{-\mu}(-L+\epsilon)$ need not be positive. Can we solve this problem by taking out $(-L+\epsilon)$ ?

Second problem is this: Even though we got for $\mu \leq 1$ this integral diverge by comparison test we get $\int_{a}^{\infty} |f(x)|dx$ diverges. It does not lead to the divergence of improper integral as stated by theorem.

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When $L \neq 0$, this limit comparison can be use to prove both absolute convergence when $\mu > 1$ and absolute divergence when $\mu \leqslant 1$.

Suppose $\lim_{x \to \infty} x^\mu f(x) = L$. Then for $\epsilon = |L|/2$ there exists $x_0 > 0$ such that when $x \geqslant x_0> 0$ we have

$$| |x^\mu f(x)| - |L|| \leqslant |x^\mu f(x) - L| \leqslant |L|/2.$$

Whence,

$$-|L|/2 \leqslant |x^\mu f(x)| - |L| \leqslant |L|/2\\ \implies |L|/2 \leqslant |x^\mu f(x)| \leqslant 3|L|/2 \\ \implies (|L|/2)x^{-\mu} \leqslant | f(x)| \leqslant (3|L|/2)x^{-\mu} $$

and the integrals of $|f(x)|$ and $x^{-\mu}$ must converge or diverge together.

The limit comparison test is often presented only for the case where $f(x) \geqslant 0$ for all $x > a$. In that case, if $\lim_{x \to \infty}x^\mu f(x) = L > 0$, then there exists $x_0$ such that when $x \geqslant x_0$ we have

$$0 < \frac{L}{2}x^{-\mu} < f(x) < \frac{3L}{2}x^{-\mu},$$

and $\displaystyle \int_a^\infty f(x) \, dx$ diverges if $\mu \leqslant 1$ (using the left inequality) and converges if if $\mu > 1$ (using the right inequality).

An example of the former case is $f(x) = \sin(1/x)$ - where $f$ is eventually positive. Since $\lim_{x \to \infty}x \sin(1/x) = 1$, then $\displaystyle \int_1^\infty \sin(1/x) \, dx$ diverges.

If $L = 0$, then for any $\epsilon > 0$ there exists $x_0$ such that when $x \geqslant x_0$ we have

$$-\epsilon x^{-\mu} < f(x) < \epsilon x^{-\mu},$$

and we can only conclude that $\displaystyle \int_a^\infty f(x) \, dx$ converges if $\mu > 1$ (using the right inequality). An example is the gamma function

$$\displaystyle \Gamma(s) = \int_0^\infty x^{s-1}e^{-x} \, dx,$$

where proof of convergence is facilitated using the test function $x^{-2}$.

If $L = 0$, then we cannot conclude divergence if $\mu \leqslant 1$ since the left lower bound is negative and the integral could converge to a positive value.

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  • $\begingroup$ So we cannot prove divergence from this theorem because If $\mu \leq 1$ left side diverge it will lead to the conclusion that integral of $|f(x)| $ do not converge.But it does not lead to the conclusion that integral of $f(x)$ diverge.Am I right..should we change the statement of theorem..? $\endgroup$ – Madhu Mar 2 '15 at 2:03
  • $\begingroup$ @Madhu: I added to the answer. Now I think you are concerned about proving divergence for a function which changes sign infinitely often and where the limit is $0$. In that case this test will not help. $\endgroup$ – RRL Mar 2 '15 at 3:28
  • $\begingroup$ okey now it is solved the statement has to modify slightly...$f(x) \geq 0$ and $L \neq 0$. $\endgroup$ – Madhu Mar 2 '15 at 16:18
  • $\begingroup$ Can you suggest any good book which discuss theorems and proofs of different types of improper integrals in university level... $\endgroup$ – Madhu Mar 2 '15 at 16:20
  • $\begingroup$ Yes, with those restrictions it is true. The usual analysis books (Rudin, Apostol, Bartle, etc.) cover improper integrals to some extent. Bartle is more extensive than the others. A Second Course in Mathematical Analysis by Burkill and Burkill gives more intricate examples. $\endgroup$ – RRL Mar 2 '15 at 16:32
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At the end of RRL's answer is the remark

If $L=0$ then we cannot conclude divergence if $\mu\le 1$ since the left lower bound is negative and the integral could converge to a positive value.

Here are counterexamples that indicate divergence and convergence are both possible for $0<\mu\le 1$, $L=0$.

  1. Divergence - $x^\mu \cdot \frac1{x \log x} \to 0 $ for every $0<\mu\le 1$, and yet $$ \int_{10}^\infty \frac1{x \log x}\, dx = \infty. $$

  2. Convergence - $x^\mu \cdot \frac1{x (\log x)^2} \to 0 $ for every $0<\mu\le 1$, and yet $$ \int_{10}^\infty \frac1{x (\log x)^2}\, dx = \frac1{\log 10}< \infty. $$ (Of course, a simpler example is $\int_0^\infty 0\, dx $, but the above example is also divergent if $\mu > 1$, so the convergence is not implied by the positive result.)

While I'm at it I'll also point out that strictly speaking, only the convergence of the Gamma function's defining integral "at infinity" is treated in RRL's answer, but it isn't too hard to deal with the singularity at 0 (if $0<s<1$.)

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A Test for the Convergence of Improper Integrals

If a function, fx is in the form cgx and has a value of zero as x approaches infinity, or infinity as x approaches zero, if at the limit xfx=0 the integral is convergent otherwise it is divergent.
Proof

Case (1) The function approaches zero as x approaches infinity Using the comparison test, the function of x must eventually be less than a convergent integral. I will compare this to ∫1/x^(1+n)dx where n is a positive number
gx/(1/x^(1+n)) = x x^n gx

As n approaches zero x x^n gx approaches x gx, and as "n" can be infinitely small but not zero the function and the comparison must have the same positive or negative slope and the absolute value of the ratio at the limit must be less than but not equal to one for the integral to be convergent. If lim/(x→∞) |x g(x)|<1 the integral is convergent

Case (2) The function approaches infinity as x approaches zero Using the comparison test, the function of x must eventually be less than a convergent integral. I will compare this to ∫1/x^(1-n)dx

Again as "n" can be infinitely small but not zero the function and the comparison must have the same positive or negative slope and the absolute value of the ratio at the limit as x approaches zero must be less than but not equal to one, for the integral to be convergent. if lim/(x→∞) |x g(x)|<1 the integral is convergent

As in both cases as the coefficient has been removed the only possible results are zero one or infinity, and as c gx =fx at the limit if the integral is convergent x fx must also equal zero.

This is somewhat similar to the mu test but takes into consideration the coefficient and displacement and can be used for vertical asymptotes replacing x with (x-a) and taking the limit as x approaches a. I have never had a failure but would like to hear of any exception.

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  • $\begingroup$ Please use MathJax. $\endgroup$ – TheSimpliFire Sep 14 '19 at 14:30

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