Does $\sum_{n=1}^\infty \frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^n}$ converge?

Question

I've been thinking about this one and I can't seem to find a way to solve it!

Does the series $$\sum_{n=1}^\infty \frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^n}$$ converge?

So far, the root test, the quotient limit test and the ratio test don't seem like they're useful. I suspect the answer comes from a direct comparison of another series but I can't seem to find one.

Any help would be appreciated!

Answer 1

I don't think it's that difficult. The individual terms seem to be approaching $1$ in the limit, so the series must diverge.

$$\frac{n^{n+1/n}}{(n+\frac1n)^n} = \frac{n^n\cdot n^{1/n}}{n^n\big(1+\frac1{n^2}\big)^n} = \frac{n^{1/n}}{\big(1+\frac1{n^2}\big)^n} \to 1.$$ Note that $\log (1+1/n^2)^n = n \log(1+1/n^2) \sim \frac 1n\to 0$.

Answer 2

Nope, $$\lim\limits_{n\rightarrow \infty} \frac{n^{n+1/n}}{(n+1/n)^n} = 1 \neq 0.$$

If $\sum_{n=0}^{\infty} \frac{n^{n+1/n}}{(n+1/n)^n}$ coverged, we would have $\lim\limits_{n\rightarrow \infty} \frac{n^{n+1/n}}{(n+1/n)^n} = 0.$

Answer 3

Notice that the $n$th term in the series is: $$\frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^{n}}=\frac{n^{n}n^{\frac{1}{n}}}{n^{n}(1+\frac{1}{n^{2}})^{n}}=\frac{n^{\frac{1}{n}}}{(1+\frac{1}{n^{2}})^{n}}$$ Taking limits as $n$ tends to $\infty$ results in $\frac{1}{1}=1$. By the Vanishing Condition, the series diverges.