Proving by definition that sequence is Cauchy

Question

I need to prove by using the definition that sequence is Cauchy. Sequence in question is $$a_n=\sum_{k=1}^n {(\arccos{1 \over k} +3k-1)\over (2k-1)^4}$$ I was confused by sum and don't know how to proceed and what to compare it with.

Answer 1

Hint. We have that $$\lim_{k\to +\infty} \frac{{\arccos(1/k) +3k-1\over (2k-1)^4}}{1/k^3}=\frac{3}{16}<1$$ which implies that there exist $N>0$ such that for $k> N$, $$0\leq {\arccos(1/k) +3k-1\over (2k-1)^4}\leq \frac{1}{k^3}$$ Hence for $n\geq m> N$, $$ 0\leq a_n-a_m\leq \sum_{k=m+1}^n\frac{1}{k^3}=b_n-b_m.$$ where $b_n:=\sum_{k=1}^{n}\frac{1}{k^3}$. Now note that $(b_n)_n$ is a convergent sequence.