Mathematics

Problem

Let

\displaystyle S=\sum_{n=1}^{\infty} \frac{1}{n(n+1)}

and let

S_N

be the

N

th partial sum. Suppose you have already calculated

S_{20}

. (a) Determine the exact value of

S

. (b) Show that the remainder after

N

terms can be written

S-S_N = \dfrac{1}{N+1}

. (c) Using your result from part (b), compute an explicit numerical upper bound for

|S-S_{20}|

and state that bound to five decimal places.