A technology startup receives a royalty payment of dollars at the end of year for

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

Problem

A technology startup receives a royalty payment of

\dfrac{3^n+1}{5^n}

dollars at the end of year

n

for

n=1,2,3,\dots

. Management is interested in the total accumulated royalty

R=\sum_{n=1}^{\infty}\dfrac{3^n+1}{5^n}

. (a) Rewrite each term

\dfrac{3^n+1}{5^n}

as a sum of two simpler fractions and state the resulting form of

R

. (b) Decide whether

R

converges or diverges. Provide a clear argument that relies on appropriate inequalities with a well–chosen benchmark series. (c) If

R

converges, find the least integer

N

such that the first

N

payments account for at least 99\% of the total accumulated royalty. Support your answer quantitatively.