## Calculus with Applications (10th Edition)

$${S_1} = \frac{1}{2},{S_2} = \frac{7}{{10}},{S_3} = \frac{{33}}{{40}},{S_4} = \frac{{403}}{{440}},{S_5} = \frac{{3041}}{{3080}}$$
\eqalign{ & {a_n} = \frac{1}{{3n - 1}} \cr & {\text{Find the first five terms of the sequence}} \cr & {a_1} = \frac{1}{{3\left( 1 \right) - 1}} = \frac{1}{2} \cr & {a_2} = \frac{1}{{3\left( 2 \right) - 1}} = \frac{1}{5} \cr & {a_3} = \frac{1}{{3\left( 3 \right) - 1}} = \frac{1}{8} \cr & {a_4} = \frac{1}{{3\left( 4 \right) - 1}} = \frac{1}{{11}} \cr & {a_5} = \frac{1}{{3\left( 5 \right) - 1}} = \frac{1}{{14}} \cr & {\text{Then by definition of partial sum}} \cr & {S_1} = {a_1} = \frac{1}{2} \cr & {S_2} = {a_1} + {a_2} = \frac{1}{2} + \frac{1}{5} = \frac{7}{{10}} \cr & {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{2} + \frac{1}{5} + \frac{1}{8} = \frac{{33}}{{40}} \cr & {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{2} + \frac{1}{5} + \frac{1}{8} + \frac{1}{{11}} = \frac{{403}}{{440}} \cr & {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{2} + \frac{1}{5} + \frac{1}{8} + \frac{1}{{11}} + \frac{1}{{14}} = \frac{{3041}}{{3080}} \cr & {\text{the partial sums are}}: \cr & {S_1} = \frac{1}{2},{S_2} = \frac{7}{{10}},{S_3} = \frac{{33}}{{40}},{S_4} = \frac{{403}}{{440}},{S_5} = \frac{{3041}}{{3080}} \cr}