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

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Answer 1 · 2014-09-11T21:33:42

$S_5=sum_{n=1}^5ln({n+1}/n)$

by writing them all out,
$=ln(2/1)+ln(3/2)+ln(4/3)+ln(5/4)+ln(6/5)$

by repeatedly applying the log property $lnx+lny=ln(xy)$ ,
$=ln(2/1cdot3/2cdot4/3cdot5/4cdot6/5)$

by cancelling common factors,
$=ln6$

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