Properties of Summations and Integrals question

[Newbie234]

Let's say we have the statement $\sum^{\infty}_{0}f(x)=\frac{\sum^{\infty}_{0}g(x)}{\sum^{\infty}_{0}h(x)}$ does this imply that
$\int^{\infty}_{0}f(x)=\frac{\int^{\infty}_{0}g(x)}{\int^{\infty}_{0}h(x)}$?

Also if $\sum^{\infty}_{0}f(x)=\sum^{\infty}_{0}g(x)$ does this imply that $f(x)=g(x)$, or just that f(x)~g(x) (asymptotically equivalent)?

Thanks.

 

[hunt_mat]

Your first statement makes no sense, what are you summing over?

 

[Newbie234]

Your first statement makes no sense, what are you summing over?

Sorry, thought it was pretty obvious, but I guess not. I didn't think it was necessary to add dx to the integrals or add that summation was over x as it was implied, and I didn't want to typeset anymore than I had to because it is new to me.

 

[micromass]

Sorry, thought it was pretty obvious, but I guess not. I didn't think it was necessary to add dx to the integrals or add that summation was over x as it was implied, and I didn't want to typeset anymore than I had to because it is new to me.

Are you summing over all the integers x, or are you summing over all the reals x? If you assume the latter, it doesn't make much sense. If you're meaning the former, then consider a function like

$$g(x)=\left\{\begin{array}{c}1/x^2~\text{if x is an integer}\\ 0~\text{otherwise}\end{array}\right.$$

that should be the basis of a counterexample for both statements.

 

[Newbie234]

Are you summing over all the integers x, or are you summing over all the reals x? If you assume the latter, it doesn't make much sense. If you're meaning the former, then consider a function like

$$g(x)=\left\{\begin{array}{c}1/x^2~\text{if x is an integer}\\ 0~\text{otherwise}\end{array}\right.$$

that should be the basis of a counterexample for both statements.

Summing over integers. Also, are these statements true for continuous functions that aren't piecewise?

 

[micromass]

Summing over integers. Also, are these statements true for continuous functions that aren't piecewise?

I don't think so. Similar counterexamples hold. Try to find a continuous version of my counterexample!

 

[Newbie234]

The first statement is false. But I am correct in assuming that the second statement implies that the infinite sums of f(x) and g(x) are asymptotically equivalent (that is, $lim_{n->\infty}\frac{\sum^{n}_{x=0}f(x)}{\sum^{n}_{x=0}g(x)}=1$).