I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 8: Testing Sequences for Convergence

Taking advantage of the theorem on the existence of a limit of monotonic bounded sequence, prove that the following sequences are convergent :
$x_n=\frac{n^2-1}{n^2}$ 

Taking advantage of the theorem on the existence of a limit of monotonic bounded sequence, prove that the following sequences are convergent :
$x_n=2+\frac{1}{2!}+\frac{1}{3!}+\dots .+\frac{1}{n!}$

Prove the existence of the limit of the sequence $y_n=a^{\frac{1}{2^n}}\left(a>1\right)$ and calculate it.

Taking advantage of the theorem on the limit of a monotonic sequence, prove the existence of a finite limit of the sequence
$x_n=1+\frac{1}{2^n}+\frac{1}{3^2}+\dots +\frac{1}{n^2}$ 

Taking advantage of the theorem on passing to the limit in inequalities, prove that
$\underset{n\to \infty }{\lim }{x}_n=1$ if $x_n=2n\left(\sqrt{n^2+1}-n\right)$

Prove that the sequence $x_1=\sqrt{a};x_2=\sqrt{a+\sqrt{a}}$, $x_3=\sqrt{a+\sqrt{a+\sqrt{a}}};\dots ;x_n=\underset{n\mathrm{radicals} }{\underbrace{\sqrt{a+\sqrt{a+\dots .+\sqrt{a}}}}}$
has the limits $b=\frac{\left(\sqrt{4a+1}+1\right)}{2}$.

Prove that the sequence with the general term $x_n=\frac{1}{3+1}+\frac{1}{3^2+2}+\dots +\frac{1}{3^n+n}$ has a finite limit.

Prove that a sequence of lengths of perimeters of regular polygon $2^n$ inscribed in a circle tends to a limit (called the length of circumference).