Which criterion do we use for the convergence of the sequences?

Question

I want to check as for the convergence the following sequences:

$$\left( \left( 1+\frac{1}{n}\right)^{n^2}\right), \left( \left( 1+\frac{1}{n^2}\right)^{n}\right), \left( \left( 1+\frac{1}{\sqrt{n}}\right)^{n}\right), \left( \left( 1+\frac{1}{2n}\right)^{n}\right).$$

Do we use the criterion that every monotone and bounded sequence converges? Or do we use an other criterion?

Answer 1

In each case, it is easy to determine whether the sequence converge or not using the fact that$$(\forall y\in\mathbb{R}):\lim_{x\to+\infty}\left(1+\frac yx\right)^x=e^y.$$

Answer 2

Hints:

There exists $\,N\in\Bbb N\; $ s.t. for $\;n>N\;$ :

$$2\le\left(1+\frac1n\right)^n\le3\implies 2^n\le\left(1+\frac1n\right)^{n^2}\le3^n$$

As before, for $\;n>M\;$ :

$$2\le\left(1+\frac1{n^2}\right)^{n^2}\le3\implies\sqrt[n]2\le\left(1+\frac1{n^2}\right)^n\le\sqrt[n]3$$

and etc. Apply the squeeze theorem and do the other cases