How could I prove these sequences are Cauchy Sequences?

Question

a) For $(s_{n})_{n\in\mathbb{N}}$, $s_{n}=\sum_{j=1}^{n}\frac{3(-1)^j+1}{2j}$

b) For $(a_{n})_{n\in\mathbb{N}}$, $a_{n}\in\mathbb{R}$, and assume that for all $n\geq N$, $N \in \mathbb{N}$ fixed, we have $|a_{n}-a_{n+1}|<3^{-n}$.

Cheers.

Answer 1

HINT: a) try to use the Leibniz criterion on this sum. If you show that the sum converges, then $(s_n)_n$ is cauchy. b) Try to make a telescoping series and make use of the triangle inequality

Update: I took up this question again because it came back to my mind and I were sceptic about (a). I think this one is little delicate: If $(s_n)_n$ was really cauchy, then the sequence would converge. But consider the following $$s_n=\sum_{j=1}^n \frac{3(-1)^j+1}{2j} = \sum_{j=1}^n \frac{(-1)^j}{j} +\frac{(-1)^j+1}{2j}.$$ If $s_n$ converges, then (since $\sum_{j=1}^n \frac{(-1)^j}{j}$ converges with Leibniz) the series $\sum_{j=1}^n \frac{(-1)^j+1}{2j}$ would converge, too. But this is false because this part is more or less a harmonic series. Therefore, $(s_n)_n$ is not cauchy.

Answer 2

For (b) you should probably use the fact:

Assuming $m\geq n$ , $\sum_{j=n}^{m-1} |a_j-a_{j+1}|\geq|a_n-a_m|$.