# Improper Integrals and Series (convergence and divergence)

1. May 23, 2013

### Lebombo

Is it safe to say when an integral has an infinite boundary $\int_n^∞ a_{n}$ and the limit yields a finite number, then the integral is said to converge.

And when a series has an upper limit of infinity $\sum_n^{∞}a_{n}$ and the limit yields a finite number, then the series is said to diverge.

Last edited: May 23, 2013
2. May 24, 2013

### SteamKing

Staff Emeritus
If the limit of an infinite series yields a finite number, the series converges. Why would you say opposite things about series and integrals?

3. May 29, 2013

### Lebombo

Oh, I made a conceptual mistake. I was thinking that if the limit of an infinite series yields a finite number, it meant that the terms of the series would "taper off" at some constant term. And that this constant term would be added up indefinitely, thus ∞ = divergent. For instance, if the limit of an infinite sequence yielded the finite number, 1/4, this meant:

a_{1}+ a_{2} + a_{3} + a_{4} + a_{5}.....1/4 + 1/4 + 1/4 + 1/4 + 1/4.... = ∞ , thus divergent

I understand now that the finite number, 1/4, in the limit of an infinite series means, that the actual sum of the series is a finite number, thus converges.

a_{1}+ a_{2} + a_{3} + a_{4} + a_{5}..... = 1/4 ,thus convergent.

Thanks for the correction.

4. May 30, 2013

### HallsofIvy

Note that "yielding an infinite number" is not the only way a series can diverge.
The series $\sum_{n= 0}^\infty (-1)^n$ is also divergent.