Convergence of $\sum\limits_{n=1}^\infty \left(\frac{2014}{n}\right)^{\frac{n}{2014}}$

The following series problem was in our last term test paper.

Prove that the following series is convergent\divergent.

$$\sum\limits_{n=1}^\infty \left(\frac{2014}{n}\right)^{\frac{n}{2014}}$$

Please give me a help. Thanks.

• What have you tried already? – Klangen Nov 28 '18 at 10:00

Notice that $\sqrt[n]{\left(\frac{2014}{n}\right)^{\frac{n}{2014}}}=\left(\frac{2014}{n}\right)^{\frac{1}{2014}}$ and $\lim \limits_{n\to \infty}\left(\frac{2014}{n}\right)^{\frac{1}{2014}}=0$.
Therefore $\lim \limits_{n\to \infty}\sqrt[n]{\left(\frac{2014}{n}\right)^{\frac{n}{2014}}}=0<1$. What does the root test say ?
For any $1>\epsilon>0$ and all $n$ sufficiently large ($n>2014$ at least), $(2014/n)^{1/2014}\leq \epsilon<1$. Apply the geometric series test and comparison test.
For $n>2014$, just compare with the series $$\sum \bigg(\frac{2014}n\bigg)^{2015/2014},$$ which converges by the integral test.
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