Test the convergence of the series $1+\frac 1 {2^2}+\frac {2^2}{3^3}+\frac {3^3}{4^4}+\cdots$

Question

A series is given by $$1+\frac 1 {2^2}+\frac {2^2}{3^3}+\frac {3^3}{4^4}+\cdots$$

My Try: I can write this series as $1+\displaystyle \sum_{n=1}^\infty \frac {n^n}{{(n+1)}^{n+1}}$. Then how should I proceed to to check whether it is convergent or not?

Answer 1

Because $$\frac{n^n}{(n+1)^n}=\left(1-\frac1{n+1}\right)^n\xrightarrow[n\to\infty]{} e^{-1}$$

the series is equivalent to the Harmonic series, hence divergent by the Limit Comparison test:

If $a_n$ denotes the $n$th term, $$a_n=\frac{n^n}{(n+1)^n}\cdot\frac1{n+1}\sim\frac{e^{-1}}{n+1}$$

Answer 2

Use $(1+\frac1n)^n\to e$ to show that your series is asymptotically equivalent to the harmonic series.