Let . Complete the following tasks to approximate to within…

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Let

S=\sum_{n=0}^{\infty}3\left(\dfrac{2}{5}\right)^{n}

. Complete the following tasks to approximate

S

to within

0.0001

. (a) Show that the difference between two successive partial sums,

|S_{N+1}-S_{N}|

, equals

3\left(\dfrac{2}{5}\right)^{N+1}

. Explain why this difference can be used as an upper bound on the absolute error

|S-S_{N}|

. (b) Determine the least integer

N

such that using

S_{N}

guarantees an approximation accurate to within

0.0001

. (c) Calculate

S_{N}

found in part (b) and record the value to five decimal places.