- #1

CGandC

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- Homework Statement
- Theorem: Given the continuous functions ## f,g : [ 1, +\infty ) \to \mathbb{R} ## that satisfy ##\lim _{x \rightarrow+\infty} \frac{f(x)}{g(x)}=L>0##. Prove the improper integrals ## \int_1^{+\infty} f(x)dx , \int_1^{+\infty} g(x)dx ## converge or diverge together.

- Relevant Equations
- * ( Integral comparison ) Let ## f,g : [ a,\omega) \to \mathbb{R} ## ( where ## \omega \in \mathbb{R} ## or ## \omega = \infty ## ). Suppose for all ## b < \omega ## it occurs that ## f,g \in \mathbb{R([a,b])} ##. Suppose ## 0 \leq f \leq g ##. If ## \int_a^\omega g < \infty ## then also ## \int_a^\omega f < \infty ##, and if ## \int_a^\omega f = \infty ## then ## \int_a^\omega g = \infty ## .

* ## f \in R([ a,b]) ## means ## f ## is Riemann-Integrable on ## [ a,b] ##

**Attempt:**

Note we must have that

## f>0 ## and ## g>0 ## from some place

or

## f<0 ## and ## g<0 ## from some place

or

## g ,f ## have the same sign in ## [ 1, +\infty) ##.

Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a sequence ## 1<x_n \to \infty ## s.t. ## \lim _{n \rightarrow+\infty} \frac{f(x_n)} {g(x_n)}<0## , a contradiction to the given ##\lim _{x \rightarrow+\infty} \frac{f(x)}{g(x)}=L>0 ##.

Suppose ## f>0 ## and ## g>0 ## from some place ( proof for ## f<0 ## and ## g < 0 ## from someplace or proof for the case where ## g,f ## have same sign in ## [1, +\infty ) ## are very similar

).

From the given ## \lim _{x \rightarrow+\infty} \frac{f(x)}{g(x)}=L>0 ## we can say that ## \forall \epsilon>0 . \exists R>0 . \forall x>R . | \frac{f(x)}{g(x)} - L | < \epsilon ##.

Note that ## -\epsilon < \frac{f(x)}{g(x)} - L < \epsilon \longrightarrow

L - \epsilon < \frac{f(x)}{g(x)} < L + \epsilon \longrightarrow (L-\epsilon)g(x) < f(x) ##.

Hence, for ## \epsilon = \frac{L}{2} ## there exists ## R>0 ## s.t. for all ## x>R ## we have that ## (L-\epsilon)g(x) = \frac{L}{2} g(x) < f(x) ##.

Suppose without loss of generality that ## R>1 ##.

Suppose that ## \int_1^{+\infty} f(x)dx ## converges and we'll show that ## \int_1^{+\infty} g(x)dx ## converges

**(**proving that if ##\int_1^{+\infty} g(x)dx## converges then ## \int_1^{+\infty} f(x)dx ## converges is similar if we look at ## f(x) < ( L + \epsilon) g(x) ##

**)**.

Under the new assumption, note that ## \int_R^{+\infty} f(x)dx > \frac{L}{2} \int_R^{+\infty} g(x) dx ##, hence from comparison test for integrals ## \int_R^{+\infty} g(x) dx ## exists.

Now, note that ## \int_1^{+\infty} g(x)dx = \int_1^{R} g(x)dx + \int_R^{+\infty} g(x)dx ##, Hence ## \int_R^{+\infty} g(x)dx = \int_1^{+\infty} g(x)dx - \int_1^{R} g(x)dx ## and since ## \int_R^{+\infty} g(x) dx ## exists then these two integrals also exist, hence ## \int_1^{+\infty} g(x)dx ## exists.

**(**After proving that if ##\int_1^{+\infty} g(x)dx## converges then ## \int_1^{+\infty} f(x)dx ## converges, we proved that ##\int_1^{+\infty} g(x)dx ~ \text{ converges} \iff \int_1^{+\infty} f(x)dx ~ \text{converges} ##. Hence we also proved ##\int_1^{+\infty} g(x)dx ~\text{ diverges} \iff \int_1^{+\infty} f(x)dx ~ \text{diverges} ##

**)**

## \square ##

**There are couple of things that I have questions about:**

1. Where have I used the fact that ##f,g ## are continuous? It seems like I proved the theorem without using this given.

( I know if a function is continuous on a bounded interval the it is Riemann Integrable on that interval, but It seems to me that under the assumption that ## \int_1^{+\infty} f(x)dx ## converges, the assumption that ##f,g ## are continuous is necessary in-order to say that ##g ## is Riemann Integrable on ## [1,\infty) ## , hence I could perform the integrable ##\int_1^{+\infty} g(x)dx ## in the inequality ## \int_R^{+\infty} f(x)dx > \frac{L}{2} \int_R^{+\infty} g(x) dx ## )

2. Was I right that the cases " ## f,g> 0 ## from some place or ##f,g< 0 ## from some place or ## g,f ## have the same sign in ## [1,+\infty) ## " are the all the possible cases?

3. I think that the proofs for the other cases ( besides " ## f,g> 0 ## from some place " ) are very similar for all the cases I have written, is that right?

4. what do you think about the proof? Is it specious? if so, then why? if not, then what could be done better?