# How do you find the 5-th partial sum of the infinite series sum_(n=1)^ooln((n+1)/n) ?

Sep 11, 2014

${S}_{5} = {\sum}_{n = 1}^{5} \ln \left(\frac{n + 1}{n}\right)$

by writing them all out,
$= \ln \left(\frac{2}{1}\right) + \ln \left(\frac{3}{2}\right) + \ln \left(\frac{4}{3}\right) + \ln \left(\frac{5}{4}\right) + \ln \left(\frac{6}{5}\right)$

by repeatedly applying the log property $\ln x + \ln y = \ln \left(x y\right)$,
$= \ln \left(\frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5}\right)$

by cancelling common factors,
$= \ln 6$