Approximation of a function as it goes to infinity

Question

For $0< H <1$. How does

\begin{align} \frac{1}{2}[(n+1)^{2H} + (n-1)^{2H}-2n^{2H}] &\approx H(2H-1)n^{2H-2} \\ & \to 0, \quad n \to \infty \end{align}

I don't understand why does the left-hand side of the equation can be approximated by the right-hand side of the equation...

Thank you in advance!

Answer 1

Hint:

By the binomial theorem,

$$(n\pm1)^a=n^a\pm an^{a-1}+\frac{a(a-1)}2n^{a-2}\pm\frac{a(a-1)(a-2)}{3!}n^{a-3}+\cdots$$

Apply it and simplify.