Answer
${s_1} = 0.2$
Work Step by Step
Write
$S = \dfrac{1}{{{1^5} + 4\cdot1}} - \dfrac{1}{{{3^5} + 4\cdot3}} + \dfrac{1}{{{5^5} + 4\cdot5}} - \dfrac{1}{{{7^5} + 4\cdot7}} + \cdot\cdot\cdot = \mathop \sum \limits_{k = 1}^\infty \dfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{{\left( {2k - 1} \right)}^5} + 4\cdot\left( {2k - 1} \right)}}$,
where ${a_k} = \dfrac{1}{{{{\left( {2k - 1} \right)}^5} + 4\cdot\left( {2k - 1} \right)}}$.
The upper bound on the absolute error by Theorem 9.6.2 is ${a_{n + 1}}$, given by Eq. (4):
$\left| {S - {s_n}} \right| \le {a_{n + 1}}$
To obtain two decimal-place accuracy, we choose a value of $n$ such that
$\left| {S - {s_n}} \right| \le {a_{n + 1}} \lt 0.005$
$\dfrac{1}{{{{\left( {2n + 1} \right)}^5} + 4\cdot\left( {2n + 1} \right)}} \lt 0.005$
${\left( {2n + 1} \right)^5} + 4\left( {2n + 1} \right) \gt 200$
Solving this inequality using a calculating utility, we obtain $n \gt 0.93$. Thus, we can take $n=1$.
Thus, the approximated sum is ${s_1}$:
${s_1} = \dfrac{1}{{{1^5} + 4\cdot1}} = \dfrac{1}{5} = 0.2$
