Answer
$s_{10}\approx 1.98828$, error $\leq 0.012$
Work Step by Step
By the power of calculator $s_{10}=\Sigma_{n=1}^{10}\frac{n}{2^{n}}\approx 1.98828$
$R_{n}=\dfrac{a_{n+1}}{a_{n}}=\dfrac{\frac{n+1}{2^{n+1}}}{\frac{n}{2^{n}}}=\frac{1}{2}+\frac{1}{2n}$
$\lim\limits_{n \to \infty}(\frac{1}{2}+\frac{1}{2n})=\frac{1}{2}$
Now, $R_{10}\leq \frac{a_{11}}{1-r_{11}}=\dfrac{a_{11}}{1-\dfrac{a_{12}}{a_{11}}}$
error $\leq 0.012$
Hence, $s_{10}\approx 1.98828$, error $\leq 0.012$
