# Limit of a sequence (TIFR 2013 problem 32)

Question:

True/False?

$$\lim_{n\to \infty } (n+1)^{1/3} -n^{1/3} = \infty$$

Hint:

Simplify the given expression.

Discussion:

We feel that $$(n+1)^{1/3}$$ goes to infinity at the same speed as $$n^{1/3}$$. So in fact, the above limit should be zero.

We make this little bit more rigorous.

$$(n+1)^{1/3} -n^{1/3} = \frac{n+1-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} }$$

$$=\frac{1}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \to 0$$ as $$n\to \infty$$.