Evaluate $lim_{n \to \infty} (n + 1)^{\frac{2}{3}} - (n)^{\frac{2}{3}}$ using the generalized binomial theorem

Question

I have already checked this question and have understood the approaches using the Product Rule for Limits, and the one using the Mean Value Theorem.

I was wondering whether it is possible to evaluate this limit using the Binomial Theorem for Rational Exponents.

After expanding, we get : $$ (n+1)^{\frac{2}{3}} - (n)^{\frac{2}{3}} = \frac{2}{3} \cdot \sqrt[3]{\frac{1}{n}} - \frac{1}{9} \cdot {n}^{\frac{-4}{3}} + \frac{4}{81} \cdot {n}^{\frac{-7}{3}} - \frac{7}{243} \cdot {n}^{\frac{-10}{3}} + \cdots \infty $$

I'm not sure how to rigorously prove that this series sum goes to $0$. Can you help me understand this? This is my first question on MSE so I apologise if I have violated any website rules. Thanks in advance!

Answer 1

Set $1/n=h$ to reach at $$\dfrac{(1+h)^{2/3}-1}{h^{2/3}}=?$$

Now use https://en.m.wikipedia.org/wiki/Binomial_series