Improper Integrals and Series (convergence and divergence)

[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.

 

[SteamKing]

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

 

[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.

 

[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.