Define the function for all real numbers such that the series converges.

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

Problem

Define the function

g(t)=\displaystyle\sum_{n=1}^{\infty} 4^{1-n}t^{n}

for all real numbers

t

such that the series converges. (a) Express

g(t)

as a rational function of

t

, clearly stating the interval of

t

for which your expression is valid. (b) Using the result from part (a), determine the exact value of

g\!\left(\dfrac{1}{2}\right)

. (c) Differentiate your closed‐form expression for

g(t)

to obtain

g'(t)

, and then compute

g'\!\left(\dfrac{1}{2}\right)

. Simplify your answer.