Answer
(a) Error: $\leq 6.4\times 10^{-5}$
(b) $s_{6}\approx 1.0173$
Work Step by Step
$\Sigma_{n=1}^{5}\frac{1}{n^{6}}=1+\frac{1}{2^{6}}+\frac{1}{3^{6}}+\frac{1}{4^{6}}+\frac{1}{5^{6}}$
$s_{6}\approx 1.0173$
error: $\leq \int_{5}^{\infty}\frac{1}{x^{6}}dx=\lim\limits_{b \to \infty}\int_{5}^{b}\frac{1}{x^{6}}dx$
$=\lim\limits_{b \to \infty}\frac{-1}{5x^{5}}|_{5}^{b}$
$=\lim\limits_{b \to \infty}\frac{-1}{5b^{5}}$
$=0+\frac{1}{15625}$
Error: $\leq 6.4\times 10^{-5}$
Hence,
(a) Error: $\leq 6.4\times 10^{-5}$
(b) $s_{6}\approx 1.0173$
