How do you determine if the series ln(1/2) + ln(1/3) + ln(3/4) + ... +ln[k/(k + 1)] + .... converges?

1 Answer
Massimiliano
Mar 17, 2015

First of all I think there is a little mistake, the second term would be:

ln(2/3) instead of ln(1/3).

So it would be:

sum_(k=1)^(+oo)ln(k/(k+1)).

We can use an important logarithmic rule that says:

ln(a/b)=lna-lnb.

So:

sum_(k=1)^(+oo)ln(k/(k+1))=ln(1/2)+ln(2/3)+ln(3/4)+...=

=ln1-ln2+ln2-ln3+ln3-ln4+...+lnk-ln(k+1)+....

The first term is zero, all the other will cancel each other, but there is always another that survive. The bigger becomes k the bigger becomes ln(k+1), the smaller becomes -ln(k+1). So the series will diverge to -oo.

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