Question

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

Answer 1

`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`