Clarification on Root Test

Question

Suppose I have $\sum_{n = 1}^{\infty} \ln(1 + \frac{1}{n})$. If I apply the root test, I get this:

$\lim_{n \to \infty}\sqrt[n]{\ln(1 + \frac{1}{n})} = 0 \lt 1$

I know the series diverges; what have I done wrong?

Answer 1

Note that for $0\leq x\leq 1$, the cancave property of $\ln(1+x)$ implies $$\ln(2)x\leq \ln(1+x)\leq x.$$ Hence $$1=\lim_{n\rightarrow+\infty}\left(\frac{\ln(2)}{n}\right)^{1/n}\leq\lim_{n\rightarrow+\infty}\sqrt[n]{\ln\left(1+\frac{1}{n}\right)}\leq \lim_{n\rightarrow+\infty}\left(\frac{1}{n}\right)^{1/n}=1$$ which means that, in this case, the root test is inconclusive.

However, in order to show that the series is divergent one of the two inequalities is quite useful: $$\sum_{n = 1}^{\infty} \ln\left(1 + \frac{1}{n}\right)\geq {\ln(2)}\sum_{n = 1}^{\infty} \frac{1}{n}.$$

Answer 2

I think $\lim\limits_{n\rightarrow+\infty}\sqrt[n]{\ln\left(1+\frac{1}{n}\right)}=1$.