$(a)$ Compute $s_{5}$ (the 5th partial sum) of $\displaystyle s=\sum_{n=1}^{\infty}\frac{19}{3 \, n^3}$ :
$s_{5} =$

Note: if giving a decimal approximation to this part, you may need to enter more digits than usual.

$(b)$ Estimate the error in using $s_{5}$ as an approximation of the sum of the series (i.e. use $\displaystyle \int_{5}^{\infty} f(x)\, dx \geq R_{5}$ ):
Estimate =

$(c)$ Use $n = 5$ and $$s_n + \int_{n+1}^{\infty} f(x)\,dx \; \leq \; s \; \leq \; s_n + \int_{n}^{\infty}f(x)\, dx$$ to find a better estimate of the sum:
$\leq s \leq$

Note give each answer accurate to (at least) five decimal places.