Answer
$${S_1} = 1,{S_2} = \frac{3}{2},{S_3} = \frac{{11}}{6},{S_4} = \frac{{25}}{{12}},{S_5} = \frac{{137}}{{60}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{n} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = 1 \cr
& {a_2} = \frac{1}{2} \cr
& {a_3} = \frac{1}{3} \cr
& {a_4} = \frac{1}{4} \cr
& {a_5} = \frac{1}{5} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} \cr
& {S_1} = 1 \cr
& {S_2} = {a_1} + {a_2} = 1 + \frac{1}{2} = \frac{3}{2} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = 1 + \frac{1}{2} + \frac{1}{3} = \frac{{11}}{6} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{{25}}{{12}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{{137}}{{60}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = 1,{S_2} = \frac{3}{2},{S_3} = \frac{{11}}{6},{S_4} = \frac{{25}}{{12}},{S_5} = \frac{{137}}{{60}} \cr} $$
