<s>[INST] Convergence of Sequence $a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$ Apply Cauchy's principle of convergence to prove that the sequence $\langle a_n\rangle$ defined by $$a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$$ is not convergent
My attempt :
consider, 
\begin{align*}
 |a_{2m}-a_{m}|&=|1+1/4+1/7+\ldots+\frac{1}{3m-2}+\frac{1}{3m+1}+\ldots+\frac{1}{6m-2}\\
&-(1+1/4+1/7+\ldots+\frac{1}{3m-2})| \\ 
 &=|\frac{1}{3m+1}+\ldots+\frac{1}{6m-2}|\end{align*}
What do i do next ? i get confused after this step
 [/INST] Note that
$$a_n=\sum_{k=1}^n\frac{1}{3k-2}>\sum_{k=1}^n\frac{1}{3k}=\frac{1}{3}\sum_{k=1}^n\frac{1}{k}.$$
Now, since $\displaystyle\sum_{k=1}^n\frac{1}{k}\to\infty$, when $n\to\infty$, we have
$$a_n\to\infty,\ \mbox{when}\ n\to\infty.$$
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