I've got to study the monotonicity of a few Euler like sequences, for example $\big(1+\frac{1}{n}\big)^{n+1}$ or $\big(1-\frac{1}{n}\big)^{n}$. Those are relatively easy to solve using either AM-GM or Bernoulli's inequality, but some of them can be solved using properties of $\big(1+\frac{1}{n}\big)^{n}$ indirectly, such as $\frac{\sqrt[n]{n!}}{n}$ (in this case via induction).
This particular exercise first requires to prove the extension of Bernoulli's inequality $$(1+x)^n>1+nx+\frac{n(n-1)}{2}x^2~\forall~n>3,~x>0$$ which is obvious from the binomial theorem.
Then, it asks to study the monotonicity of the sequence $$x_n=\bigg(1+\frac{1}{n}\bigg)^{n+\frac{1}{2}},~n>3$$
Trying to prove that $x_{n-1}>x_n$, we arrive at the following inequality: $$\bigg(1+\frac{1}{n^2-1}\bigg)^{2n}>1+\frac{2}{n-1}$$ which I'm having a hard time to prove using the first part of the exercise.
I've also tried other methods, but I can't seem to figure it out. Please, I need a little help.
