Consider the sequence $x_n=\sum_{j=1}^{n}\frac{1}{j}$ and $y_n=\sum_{j=1}^{n}\frac{1}{j^2}$

Question

Consider the sequence

$\begin{align} & {{x}_{n}}=\sum\limits_{j=1}^{n}{\frac{1}{j}} \\ & {{y}_{n}}=\sum\limits_{j=1}^{n}{\frac{1}{{{j}^{2}}}} \\ \end{align}$

Then

$\{{{x}_{n}}\}$ is Cauchy but $\{{{y}_{n}}\}$is not.

what I tried to show that $x_n$ is convergent by monotonic bounded. But I could not show that $x_n$ is bounded. Dont know about $y_n$

Answer 1

To prove $\{y_n\}$ is bounded, notice that $1/n^2<1/n(n-1)$ for $n\geq 2.$

So, we have that

$$\sum_1^m \frac{1}{n^2}<1+\sum_2^m \frac{1}{n(n-1)}\leq 1+1-\frac{1}{m}<2.$$

Answer 2

The first isn't Cauchy. Hint: $$x_{2n} - x_n = \frac 1{n+1} + \cdots + \frac 1{2n}\ge\cdots$$ Can you continue?