The sum $\sum_{k=1}^{\infty} \frac{1}{k^3 + 6k^2 + 8k}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a + b$?
$$\textbf{(A) } 89 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 102 \qquad \textbf{(D) } 107 \qquad \textbf{(E) } 129$$
Difficulty: 61/100 - An intimidating telescoping series problem which requires somewhat advanced concepts.
Core Concepts: partial fractions, telescoping series, infinite sums
Challenges: Competitors with no experience in factoring or decomposing expressions will have a lot of trouble making the seemingly random expression more workable.
Techniques: Rewriting the expression using partial fraction decomposition and cancelling out common terms.
Error-prone Steps: Many computation errors are possible for competitors who used a more crude method of evaluating the telescoping series.
Ideal Time:
Experienced: ≤ 4 min
Intermediate: ≤ 9 min
Beginner: ≤ 15 min if solved at all