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.

score 9 · Answer 1 · answered Oct 30 '14 at 23:57

9

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

answered Oct 30 '14 at 23:57

Kim Jong Un

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I am always surprised : the most obvious thing to be checked for the study of a series convergence is almost never checked. ;-) – Olórin Oct 30 '14 at 23:59
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Ugh this is so awkward haha – dean_mi12 Oct 31 '14 at 00:01

score 6 · Accepted Answer · answered Oct 30 '14 at 23:56

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

answered Oct 30 '14 at 23:56

Przemysław Scherwentke

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Doesn't this also work for $n > e$ ? – dean_mi12 Oct 31 '14 at 00:00
@dean_mi12 For $n\in\mathbb{N}$ this is the same. – Przemysław Scherwentke Oct 31 '14 at 00:03
Oh sorry, obviously, forgot that $n$ is a natural number – dean_mi12 Oct 31 '14 at 00:04

score 5 · Answer 3 · answered Oct 31 '14 at 00:14

5

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.

answered Oct 31 '14 at 00:14

David

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Proving the divergence of a series through a better method

3 Answers3