lecture_11

Thu Jun 21 09:00:23 CDT 2012

DNHI

Show that [\sum\limits_{n = 1}^{\infty}{ \frac{ 1}{ n(n + 1)} = 1]

• [\sum\limits_{n = 1}^{\infty}{ \frac{ 1}{ n(n + 1)} = \sum\limits_{n = 1}^{\infty}{ \left(\frac{ 1}{ n} = \frac{ 1}{ n+1}\right) = 1]
  • telescoping series

Another proof

If [0 < a_i < 1], show that

[\sum\limits_{i = 1}^{\infty}{ a_i \prod_{j = 1}^{i - 1}{ (1 - a_j)}} + \prod\limits_{i = 0}^{infty}{ (1 - a_i) = 1]