Let be a real number

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

Problem

Let

k

be a real number. Define two infinite series depending on

k

\displaystyle S(k)=\sum_{n=0}^{\infty} k\left(\dfrac13\right)^{n}, \qquad T=\sum_{n=0}^{\infty} 4\left(\dfrac14\right)^{n}.

The inequality

S(k)>T

will hold for some, all, or none of the real numbers

k

. (1) Determine the exact value of the series

T

. Show all work. (2) Express

S(k)

as a simplified function of

k

. (3) Solve the inequality

S(k)>T

for

k

, and give the solution set in interval notation.