Answer
$${S_1} = 1,{S_2} = \frac{4}{3},{S_3} = \frac{{23}}{{15}},{S_4} = \frac{{176}}{{105}},{S_5} = \frac{{563}}{{315}}$$
Work Step by Step
$$\eqalign{
& {a_n} = \frac{1}{{2n - 1}} \cr
& {\text{Find the first five terms of the sequence}} \cr
& {a_1} = \frac{1}{{2\left( 1 \right) - 1}} = 1 \cr
& {a_2} = \frac{1}{{2\left( 2 \right) - 1}} = \frac{1}{3} \cr
& {a_3} = \frac{1}{{2\left( 3 \right) - 1}} = \frac{1}{5} \cr
& {a_4} = \frac{1}{{2\left( 4 \right) - 1}} = \frac{1}{7} \cr
& {a_5} = \frac{1}{{2\left( 5 \right) - 1}} = \frac{1}{9} \cr
& {\text{Then by definition of partial sum}} \cr
& {S_1} = {a_1} = 1 \cr
& {S_2} = {a_1} + {a_2} = 1 + \frac{1}{3} = \frac{4}{3} \cr
& {S_3} = {a_1} + {a_2} + {a_3} = 1 + \frac{1}{3} + \frac{1}{5} = \frac{{23}}{{15}} \cr
& {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} = \frac{{176}}{{105}} \cr
& {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} = \frac{{563}}{{315}} \cr
& {\text{the partial sums are}}: \cr
& {S_1} = 1,{S_2} = \frac{4}{3},{S_3} = \frac{{23}}{{15}},{S_4} = \frac{{176}}{{105}},{S_5} = \frac{{563}}{{315}} \cr} $$
