Question

If an infinite series of f(x) converges if the top value tends to infinity, then does its integral from 1 to infinity always converge, and vise-versa?

Answer 1

An example:\[\int\limits_{1}^{\infty}\frac{1}{x^2}dx\text{....has a finite value, as does...}\sum_{1}^{\infty}\frac{1}{x^2}\]\[\int\limits_{1}^{\infty}\frac{1}{x}dx\text{....hasn't a finite value, as does \not...}\sum_{1}^{\infty}\frac{1}{x}\]

Answer 2

Or, worded another way: is the convergence of an integral sign of the convergence of its infinite series?

Answer 3

No, that's the weird thing that caused me to answer this question (I'm sure of that fact, but I'm not certain why...).

Answer 4

http://math.stackexchange.com/questions/5115/why-does-1-x-diverge