Consider the series

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider the series

\displaystyle \sum_{n=2}^{\infty} \frac{3}{5^{n}}

. Let

S

denote the exact sum of the series and

S_N

the

N

th partial sum (beginning with

n=2

). A scientific computation requires an approximation of

S

that is accurate to within

5\times10^{-6}

. (a) Find a closed-form expression for

S

. (b) Derive an explicit formula for

S-S_N

in terms of

N

. (c) Determine the smallest integer

N

for which

|S-S_N|<5\times10^{-6}

, and justify that your choice of

N

meets the stated accuracy requirement.