<s>[INST] I need help with a limit problem I need help with the following limit:
$\lim_{n \to \infty}\left [ (n^2+1)\left ( \left ( 1+\frac{2}{n^2} \right )^a - 1 \right ) \right ],a\in \mathbb{N}$
I am completely lost.
 [/INST] Let us consider $$A= (n^2+1)\left ( \left ( 1+\frac{2}{n^2} \right )^a - 1 \right )$$ In a first time, let us focus on $(1+x)^a$ and use the generalized binomial theorem for generating a Taylor-like series when $x$ is small. So $$(1+x)^a=1+a x+\frac{1}{2} (a-1) a x^2+\frac{1}{6} (a-2) (a-1) a x^3+O\left(x^4\right)$$ Now, replace $x$ by $\frac{2}{n^2}$ $$\left ( 1+\frac{2}{n^2} \right )^a - 1 =\frac{2 a}{n^2}+\frac{2 (a-1) a}{n^4}+\frac{4 (a-2) (a-1) a}{3 n^6}+\cdots$$ Multiply by $(n^2+1)$, expand and simplify; we arrive to $$A=2 a+\frac{2 a^2}{n^2}+\frac{2 \left(2 a^3-3 a^2+a\right)}{3
   n^4}+O\left(\left(\frac{1}{n}\right)^5\right)$$ which shows the limit and how it is approached.
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