does this series diverges or converges? $$\sum _{n=0}^{\infty }\:\frac{1}{1+\ln\left(n\right)}$$
my opinion is that the series diverges the summation gets more and more, But my book says ( the series is neither divergent or convergent )
is it divergent or convergent? and why?
 A: The $n=0$ term is undefined since $\ln(0)$ is undefined. Because of this issue, the answer in the book is what is most proper as an answer.
If the summation started at $n=1$, you're answer is correct, though your reasoning needs some improvement. One way to argue the divergence is to note that $1+\ln(n)\leq n$ for all $n\geq1$, which gives the equivalent inequality $\frac{1}{1+\ln(n)}\geq\frac{1}{n}$. From this, we can apply the comparison test to the series (comparing the series to $\sum \frac1n$) and conclude divergence of the original series since this latter series is the well-known harmonic series.
A: Just observe that for $n\ge 1$ you have that $\frac{1}{1+\ln n} > \frac 1n$ and $\sum_{n=1}^{\infty} \frac 1n$ diverges. This implies that your series also diverges.
I am talking about the series $\sum_{n=1}^{\infty} \frac{1}{1+ \ln n}$, since for $n=0$ the expression is not defined (although you could argue that the first term could be set to $\displaystyle 0 = \lim_{x\to 0} \frac{1}{1+\ln x}$. 
