Given a sequence ${x_n}$ = $sqrt(1)$ , $-sqrt(1)$,$sqrt(2)$,$-sqrt(2)$...

Given a sequence ${x_n}$ = $\sqrt{1}$ , $-\sqrt{1}$,$\sqrt{2}$,$-\sqrt{2}$...

If $y_n$ = {$x_1$ + $x_2$ + $x_3$ + $x_4$ + ... $x_n$}$1\over n$

Then sequence $y_n$ is

1.$Monotonic$

2.NOT bounded

3.bounded but not convergent (This is correct )

4.convergent

My attempt :I noticed a couple of things here

${x_n}$ is not convergent as two subsequences are convergent to different limits.

The terms of $y_n$ are $1$,$\sqrt(1)$/2,$\sqrt(2)$/3,

So ${y_n}$ seems to go to zero for large n and thus convergent ,but im not sure regd this . Can any1 help it out? Thanks

Answer

HINT: $y_{2k}=0$, $y_{2k+1}=\sqrt{k+1}/(2k+1)$.