Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - Chapter Review - Review Exercises - Page 662: 40

Answer

$${S_1} = \frac{1}{{12}},{S_2} = \frac{2}{{15}},{S_3} = \frac{1}{6},{S_4} = \frac{4}{{21}},{S_5} = \frac{5}{{24}}$$

Work Step by Step

$$\eqalign{ & {a_n} = \frac{1}{{\left( {n + 2} \right)\left( {n + 3} \right)}} \cr & {\text{Find the first five terms of the sequence}} \cr & {a_1} = \frac{1}{{\left( {1 + 2} \right)\left( {1 + 3} \right)}} = \frac{1}{{12}} \cr & {a_2} = \frac{1}{{\left( {2 + 2} \right)\left( {2 + 3} \right)}} = \frac{1}{{20}} \cr & {a_3} = \frac{1}{{\left( {3 + 2} \right)\left( {3 + 3} \right)}} = \frac{1}{{30}} \cr & {a_4} = \frac{1}{{\left( {4 + 2} \right)\left( {4 + 3} \right)}} = \frac{1}{{42}} \cr & {a_5} = \frac{1}{{\left( {5 + 2} \right)\left( {5 + 3} \right)}} = \frac{1}{{56}} \cr & {\text{Then by definition of partial sum}} \cr & {S_1} = {a_1} = \frac{1}{{12}} \cr & {S_2} = {a_1} + {a_2} = \frac{1}{{12}} + \frac{1}{{20}} = \frac{2}{{15}} \cr & {S_3} = {a_1} + {a_2} + {a_3} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} = \frac{1}{6} \cr & {S_4} = {a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} = \frac{4}{{21}} \cr & {S_5} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} = \frac{5}{{24}} \cr & {\text{the partial sums are}}: \cr & {S_1} = \frac{1}{{12}},{S_2} = \frac{2}{{15}},{S_3} = \frac{1}{6},{S_4} = \frac{4}{{21}},{S_5} = \frac{5}{{24}} \cr} $$
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