Consider the geometric series

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

\displaystyle \sum_{k=0}^{\infty} \left(\frac14\right)^k

. Throughout, let

S

denote the exact value of the infinite series and let

S_N

denote the

N

th partial sum. (a) Calculate

S

. (b) Determine the least integer

N

such that

|S-S_N|<0.001

. (c) Verify that your value of

N

indeed achieves the required accuracy by computing

|S-S_N|

to four decimal places.