Calm. Think it through.
f(k) can't be continuous.
f(x) must be integrable - Riemann Integrable.
Does that say anything about continuity?
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]