Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent 
Show that the following sum is divergent
  $$\sum_{n=1}^{\infty}\ln\left(1+\frac1n\right)$$

I thought to do this using Taylor series using the fact that
$$
\ln\left(1+\frac1n\right)=\frac1n+O\left(\frac1{n^2}\right)
$$
Which then makes it clear that
$$
\sum_{n=1}^{\infty}\ln\left(1+\frac1n\right)\sim \sum_{n=1}^{\infty}\frac1n\longrightarrow \infty
$$
But I feel like I overcomplicated the problem and would be interested to see some other solutions. Also, would taylor series be the way you would see that this diverges if you were not told?

You can also use the Abel's summation $$\sum_{n=1}^{N}\log\left(1+\frac{1}{n}\right)=\sum_{n=1}^{N}1\cdot\log\left(1+\frac{1}{n}\right)=N\log\left(1+\frac{1}{N}\right)+\int_{1}^{N}\frac{\left\lfloor t\right\rfloor }{t\left(t+1\right)}dt
 $$ where $\left\lfloor t\right\rfloor 
 $ is the floor function. Since $\left\lfloor t\right\rfloor =t+O\left(1\right)
 $ we have $$\sum_{n=1}^{N}\log\left(1+\frac{1}{n}\right)=N\log\left(1+\frac{1}{N}\right)+\log\left(N+1\right)+O\left(1\right)
 $$ and taking $N\rightarrow\infty$ we can see that the series diverges.
