Integral test:if f is continuous(?), decreasing, and POSITIVE on [n,∞), then

∞

∑ f(k) converges if and only if

k=n

∞

∫ f(x) dx converges.

n

To apply the integral test, does f have to be continuous?In my first year calculus text book, the integral test is as stated above; f is assumed to be continuous. But in my other textbook, there is no such assumption. The same for wikipedia (http://en.wikipedia.org/wiki/Integra...nt_of_the_test), it looks like there is no assumption of continuity in the integral test...how come?? Is it wrong?? Or maybe there is a typo?

What is the correct statement of the integral test? Do we need to assume continuity?

Also, I was thinking of applying integral test to determine the convergence/divergence of

∞

∑ 1 / [n log n log(log n)]

n=2

Let f(x)=1 / [x log x log(log x)]

Then f(2)<0, but the integral test requires f to be positive so I think I should start the integral at 3 and look at

∞

∫ f(x) dx.

3

But for improper integrals,

∞

∫ f(x) dx

k

Does the value of "k" here have any effect on the convergence/divergence of the improper integral?

Is it ever possible that, for example,

∞

∫ f(x) dx converges

8

∞

∫ f(x) dx diverges ?

5

Thank you for clarifying!

[also under discussion in math links forum]