Find $\sum ^{999}_{n=1}\dfrac {1}{\sqrt [3] {n^{2}+2n+1}+\sqrt [3] {n^{2}+n}+\sqrt [3] {n^{2}}}$

Question

How to find this summation $\sum ^{999}_{n=1}\dfrac {1}{\sqrt [3] {n^{2}+2n+1}+\sqrt [3] {n^{2}+n}+\sqrt [3] {n^{2}}}$

Answer 1

HINT

Notice that $a^3-b^3= (a-b)(a^2+ab+b^2)$.

This gives $$ \dfrac {1}{\sqrt [3] {n^{2}+2n+1}+\sqrt [3] {n^{2}+n}+\sqrt [3] {n^{2}}}=\frac{\sqrt [3]{n+1}-\sqrt [3]n}{n+1 -n} $$