Negation- Cauchy sequence

Question

Use negation of the definition of Cauchy sequence to prove that the sequence

$$x_n = \sum_{k=0}^n \frac1{3k+2}$$

is not a Cauchy sequence.

Answer 1

Let us take $\epsilon=1/ 2$,now take any $N \in \mathbb N$ , Now consider the tail , $\sum_{k=n+1}^{m} \frac{1}{3k+2}$,where $m>n\geq N$. Now for $n=N$ and $m=2N$,the sum $\sum_{k=n+1}^{m} \frac{1}{3k+2}$=$\sum_{k=N+1}^{2N}\frac{1}{3k+2}>\sum_{k=N+1}^{2N} \frac{1}{3(2N)+2}=\sum_{k=N+1}^{2N} \frac{1}{6N+2}=\frac{1}{6N+2}\sum_{k=N+1}^{2N} 1=\frac{1}{6N+2}\frac{N(3N+1)}{2}=\frac{N}{2}\geq\frac{1}{2}$.So we have an $\epsilon$ viz $1/2$ for which no $N$ works as form any $N$ we can find some $m>n\geq N$ such that $|x_m-x_n|\geq \epsilon$.