Homework Statement
Show that this sum is convergent if and only if k>1 :
\sum_{n=1}^{\infty}\frac{1}{n\log{n}\left(\log{\log{n}}\right)^k}The Attempt at a Solution
I've applied the Cauchy condensation test, giving:
\frac{2^n}{2^n\log{2^n}\left(\log{\log{2^n}}\right)^k}...
I have an answer now, basically because a_n\rightarrow a and f is continuous then f(a_n)\rightarrow f(a), so there exists n bigger than N such that |f(a_n)-f(a)|<\epsilon so then just choose \epsilon = f(a)/2 do some rearranging and the inequality pops out.
In other words, f(a_n) is a...
Homework Statement
Given a convergent sequence:
a_n \rightarrow a
and a continuous function:
f:\mathbb{R}\rightarrow\mathbb{R}
show that there exists an N\in\mathbb{N} such that \forall n>N:
f(a_n)\geq\frac{f(a)}{2}
Homework Equations
Usual definitions for limit of a sequence...