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If I have the following expression:

$\displaystyle\int_{0}^{\infty} f(x) dx = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^3}$

Can I then deduce that $\int_{0}^{\infty} f(x) dx$ converges, because right hand side does?

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2 Answers 2

up vote 1 down vote accepted

Well...yes, of course! What you wrote says the integral equals a number , so yes: this means the integral converges.

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Thank you! But isn't there a difference between the statement "the sum converges to a number" and "the sum is a number"? –  characters May 28 '12 at 15:57
    
Oops Sorry. I see :) I'm just a little confused. –  characters May 28 '12 at 16:00

If I were to say, for example, for a finite $C$

$$\int^{\infty}_{0}f(x)\, dx=C < \infty \implies \int^{\infty}_{0}f(x)\, dx < \infty$$

Thus

$$\displaystyle\int_{0}^{\infty} f(x) \,dx=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^3}=\zeta(3)$$

So the integral converges! You even know what the integral converges to.

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