Proving the divergence of a series through a better method

Question

Can anyone suggest an elegant proof for the divergence of the series: $$ \sum\limits_{n = 1}^\infty {7^{\ln n} } $$

I already solved it using the Raabe-Duhamel test, but I would like to see something prettier.

Answer 1

For $n\geq3$ we have $7^{\ln n}>7$.

Answer 2

Note that $\lim_n 7^{\log n}\neq 0$. (In fact, it is $+\infty$.)

Answer 3

In fact, by using properties of logarithms, the summand is just a power of $n$, namely $$7^{\ln n}=n^{\ln7}\ .$$ and since the power is greater than or equal to $-1$ (definitely!), the series diverges.