Find if this series converges and if so find its value

Question

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is decreasing. A step by step would be great since i am new to this topic.

Answer 1

$$\lim_{n\to \infty }(\ln(n+2)-\ln 2)=\lim_{n\to \infty }\ln(1+\frac n2)=+\infty$$

Answer 2

Hint: $$\ln \left(\dfrac{k+2}{k+1}\right)=\ln (k+2)-\ln(k+1)$$ Now if you expand the sum (as you already have) you will see that this is a telescoping series.

Answer 3

Look at the partial sum: $$ S_n = \sum_{k=1}^n \log\left(\frac{k+2}{k+1}\right) = \left(\sum_{k=3}^{n+2} \log k\right) - \left(\sum_{k=2}^{n+1} \log k\right) = \log(n+2) - \log 2. $$ Then, $$ \lim_{n\to\infty} S_n = \lim_{n\to\infty} \log(n+2) - \log 2 = +\infty. $$