Infinite Series Summation 1
Compute the sum
\[\sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{1}{(i + j + 1)(i + j + 2)(i + j + 3)(i + j + 4)(i + j + 5)(i + j + 6)(i + j + 7)}.\]
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
First, we can write
\begin{align*}
&\frac{1}{(i + j + 1)(i + j + 2) \dotsm (i + j + 6)(i + j + 7)} \\
&= \frac{1}{6} \cdot \frac{(i + j + 7) - (i + j + 1)}{(i + j + 1)(i + j + 2) \dotsm (i + j + 6)(i + j + 7)} \\
&= \frac{1}{6} \left( \frac{1}{(i + j + 1)(i + j + 2) \dotsm (i + j + 6)} - \frac{1}{(i + j + 2) \dotsm (i + j + 6)(i + j + 7)} \right).\end{align*}Thus, the following sum telescopes:
\begin{align*}
&\sum_{j = 0}^\infty \frac{1}{(i + j + 1)(i + j + 2) \dotsm (i + j + 6)(i + j + 7)} \\
&= \sum_{j = 0}^\infty \frac{1}{6} \left( \frac{1}{(i + j + 1)(i + j + 2) \dotsm (i + j + 6)} - \frac{1}{(i + j + 2) \dotsm (i + j + 6)(i + j + 7)} \right) \\
&= \frac{1}{6} \left( \frac{1}{(i + 1) \dotsm (i + 6)} - \frac{1}{(i + 2) \dotsm (i + 7)} \right) \\
&\quad + \frac{1}{6} \left( \frac{1}{(i + 2) \dotsm (i + 7)} - \frac{1}{(i + 3) \dotsm (i + 8)} \right) \\
&\quad + \frac{1}{6} \left( \frac{1}{(i + 3) \dotsm (i + 8)} - \frac{1}{(i + 4) \dotsm (i + 9)} \right) +\dotsb \\
&= \frac{1}{6 (i + 1)(i + 2) \dotsm (i + 5)(i + 6)}.\end{align*}We can then write
\begin{align*}
&\frac{1}{6 (i + 1)(i + 2) \dotsm (i + 5)(i + 6)} \\
&= \frac{1}{5} \cdot \frac{(i + 6) - (i + 1)}{6 (i + 1)(i + 2) \dotsm (i + 5)(i + 6)} \\
&= \frac{1}{30} \left( \frac{1}{(i + 1)(i + 2)(i + 3)(i + 4)(i + 5)} - \frac{1}{(i + 2)(i + 3)(i + 4)(i + 5)(i + 6)} \right).\end{align*}We obtain another telescoping sum:
\begin{align*}
&\sum_{i = 0}^\infty \frac{1}{6 (i + 1)(i + 2) \dotsm (i + 5)(i + 6)} \\
&= \sum_{i = 0}^\infty \frac{1}{30} \left( \frac{1}{(i + 1)(i + 2)(i + 3)(i + 4)(i + 5)} - \frac{1}{(i + 2)(i + 3)(i + 4)(i + 5)(i + 6)} \right) \\
&= \frac{1}{30} \left( \frac{1}{(1)(2)(3)(4)(5)} - \frac{1}{(2)(3)(4)(5)(6)} \right) \\
&\quad + \frac{1}{30} \left( \frac{1}{(2)(3)(4)(5)(6)} - \frac{1}{(3)(4)(5)(6)(7)} \right) \\
&\quad + \frac{1}{30} \left( \frac{1}{(3)(4)(5)(6)(7)} - \frac{1}{(4)(5)(6)(7)(8)} \right) + \dotsb \\
&= \frac{1}{30} \cdot \frac{1}{(1)(2)(3)(4)(5)} = \boxed{\frac{1}{3600}}.\end{align*}