Evaluate the infinite series .

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

Problem

Evaluate the infinite series

T=\displaystyle\sum_{k=1}^{\infty}\Bigl[\left(\dfrac{2}{5}\right)^{k}-\left(\dfrac{3}{10}\right)^{k}\Bigr]

. (a) Demonstrate that each of the two component series in the definition of

T

converges. (b) Determine the exact value of

T

in simplest fractional form. (c) Let

R_N

denote the remainder obtained by truncating the series after the

N

-th term. Find the least integer

N

such that

|R_N|<10^{-4}

, and justify your answer.