Let $a _ { k } = f ( k )$

Let $a _ { k } = f ( k )$ , where $f$ is a continuous, positive, and decreasing function on $[ n , \infty )$ , and suppose that $\sum _ { k = 1 } ^ { \infty } a _ { k }$ is convergent. Defining $R _ { n } = S - S _ { n }$ , where $S = \sum _ { n = 1 } ^ { \infty } a _ { n }$ and $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ , we have that $\int _ { n + 1 } ^ { \infty } f ( x ) d x \leq R _ { n } \leq \int _ { n } ^ { \infty } f ( x ) d x$ . Find the maximum error if the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } }$ is approximated by $S _ { 40 }$ .