Telescoping series

Problem 7.325 · easy

1. \[ \frac{6}{n^{2} + 8 n + 15} = - \frac{3}{n + 5} + \frac{3}{n + 3} \]
   Partial fractions: the term is a difference.✓ Proved
2. In the N-th partial sum every negative piece cancels a positive piece 2 terms later; only the first positive and last negative pieces survive.
   Reviewed
3. So S_N = 27/20 - 3/(N + 5) - 3/(N + 4).
   Reviewed
4. \[ \lim_{k \to \infty}\left(\frac{27}{20} - \frac{3}{k + 5} - \frac{3}{k + 4}\right) = \frac{27}{20} \]
   Let N → ∞.✓ Proved