Does the series $\sum_\limits{n=1}^{\infty}\frac{(n+1)^a-n^a}{(n+1)^a} \ n\in \mathbb N, a>0$ always diverge?

Question

Does the following series always diverge?

$$\sum_\limits{n=1}^{\infty}\frac{(n+1)^a-n^a}{(n+1)^a} \ n\in \mathbb N, a>0$$

Answer 1

Note that

$$(n+1)^a-n^a=n^a(1+1/n)^a-n^a=n^a(1+a/n+a(a-1)/n^2+...)-n^a=an^{a-1}+a(a-1)n^{a-2}+...$$

then

$$\frac{((n+1)^a-n^a)}{(n+1)^a}\sim \frac a n+a(a-1)n^{-2}+...$$

therefore the given series diverges by limit comparison test with $\sum \frac1n$.

Answer 2

By the mean value theorem, $(n+1)^a-n^a =ax^{a-1}$ where $n < x < n>1$. Therefore each term is at least $an^{a-1}/(n+1)^a \approx a/n$ since $(1+1/n)^a \to 1$. The sum of these diverges.