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

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Sep 11, 2014

#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#

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