summation of series by telescoping series method (feedback needed)

Question

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so Please see also for the errors.

Answer 1

What you have done is correct. Now it is straightforward that $$\lim_{n \to \infty} -\ln 2 + \ln(n+2)= -\ln 2 + \lim_{n \to \infty}\ln(n+2)=+\infty$$ since $\ln$ is a monotone increasing function.

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If you need to prove that $\ln n$ is unbounded you need the following: $$\ln' n=\dfrac{1}{n}>0$$ so that $\ln$ is monotone increasing. Moreover $\ln 2>\ln 1=0$. Now, take $M\in \mathbb R$, arbitrarily large. Then there exists $m \in \mathbb N$ such that $$M<m \ln 2$$ (since $\ln 2>0$) or equivalently $M< \ln 2^m$. Therefore for any $n>2^m$ you have that $$M<\ln 2^m <\ln n$$ from which you can conclude that $\ln n$ is unbounded since $M$ was arbitrarily large.