Given the following series: ∑n=1∞(n+13−n−13)α where α∈R. Does the series converge or diverge?

Answered: "Given the following series: ∑n=1∞(n+13−n−13)α where α∈R. Does..."

Jaylin Clements

2022-03-26

Given the following series:

\sum_{n = 1}^{\infty} {(\sqrt[3]{n + 1} - \sqrt[3]{n - 1})}^{α}

where

α \in R

. Does the series converge or diverge?

Jazlyn Mitchell

Beginner2022-03-27Added 14 answers

The formula

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

actually can be quite helpful. Note that it implies that

\sqrt[3]{n + 1} - \sqrt[3]{n - 1} = \frac{2}{{(n + 1)}^{\frac{2}{3}} + {(n^{2} - 1)}^{\frac{1}{3}} + {(n - 1)}^{\frac{2}{3}}}

For sufficiently large n (n>2 will do for sure, and you can get tighter bounds by going further), that denominator is squeezed between

\frac{3}{2} n^{\frac{2}{3}}

and

6 n^{\frac{2}{3}}

, and therefore the convergence question is the same as for the series

\sum_{n = 1}^{\infty} n^{- 2 \frac{α}{3}}

Do you have a similar question?

Recalculate according to your conditions!