AP

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

Problem

Let

S_n=\displaystyle \sum_{k=1}^{n} \frac{k}{3^{k}}

. (a) Show algebraically that

3S_n-\dfrac{1}{3}-S_n

can be written as a finite series whose terms simplify considerably. Then obtain an explicit formula for

S_n

that contains no summation symbol. (b) Using your formula from part (a), determine

\displaystyle \lim_{n\to\infty} S_n

or explain why the limit fails to exist. (c) Decide whether the infinite series

\displaystyle \sum_{k=1}^{\infty} \frac{k}{3^{k}}

converges or diverges. Support your conclusion with reasoning grounded in your earlier work.