MHB Finding Convergence of Series: $\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]$

[Petrus]

Hello MHB,
I am pretty new with this serie I am supposed to find convergent or divergent.
$$\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]$$

progress:
$$\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]= \sum_{k=1}^\infty [\ln(\frac{k+1}{k})] = \sum_{k=1}^\infty[\ln(k+1)-\ln(k)]$$ so we got that
$$\lim_{n->\infty}(\ln(2)-\ln(1))+$$$$(\ln(3)-\ln(2))+...+(\ln(n+1)-\ln(n))$$
and this is where I am stuck

Regards,
$$|\pi\rangle$$

 

[mathbalarka]

Re: serie

You've done fine up until this point. Consider the first few terms you have written :

$$\ln(2) - \ln(1) + \ln(3) - \ln(2) + \cdots$$

Do you see that $\ln(2)$ cancels out? Can you, similarly, show that some further terms cancel outs too? If so, can you determine what is finally left?

Spoiler

Keyword : Telescoping series

 

[Petrus]

Re: serie

You've done fine up until this point. Consider the first few terms you have written :

$$\ln(2) - \ln(1) + \ln(3) - \ln(2) + \cdots$$

Do you see that $\ln(2)$ cancels out? Can you, similarly, show that some further terms cancel outs too? If so, can you determine what is finally left?

Spoiler

Keyword : Telescoping series

EDIT : This can be shown in another way though rather easily by noting that $\log (1 + 1/k) \geq \log(1/k)$.

Ohh I see it was infront of my eyes... I did not even checking those will cancel out...! Thanks a lot now I got it!
We got left
$$\lim_{n->\infty}\ln(n+1)-\ln(1)=\infty$$
Thanks for taking your time! Guess I need to wake up!:P

Regards,
$$|\pi\rangle$$