Show that

$\rm ( 1 + \frac {1}{1!} + \frac {1}{2!} + \frac {1}{3!} + \dots )( 1 - \frac {1}{1!} + \frac {1}{2!} - \frac {1}{3!}+ \dots ) = 1 $

Show that

$\rm ( 1 + \frac {1}{2!} + \frac {1}{4!} + \dots )^2 - ( 1 + \frac {1}{3!} + \frac {1}{5!} + \dots )^2 = 1 $

Show that

$\rm\frac {2}{1!} + \frac {4}{3!} + \frac {6}{5!} + \dots $ to $\rm \infty = e $

Show that

$\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 3}{3!} + \frac {1 + 2 + 3+ 4}{4!} + \dots = \frac {3e}{2} $

Show that 

$\frac {1 + \frac{1}{2!} + \frac {1}{4!} + \frac {1}{6!} + \dots }{1 + \frac{1}{3!} + \frac {1}{5!} + \frac {1}{7!} + \dots } = \frac {e^2 + 1}{e^2 - 1}$ 

Show that

$\rm \frac {1}{1!} + \frac {1 + 3}{2!} + \frac {1 + 3 + 5}{3!} + \frac {1 + 3 + 5 + 7}{4!} + \dots = 2e $

Sum to infinity 

$\rm \frac {1.2}{1!} + \frac {2.3}{2!} + \frac {3.4}{3!} + \dots $

Sum to infinity 

$\rm 1 + \frac {3}{1!} + \frac {5}{2!} + \frac {7}{3!} + \dots $

Sum to infinity 

$(\rm 1 + \frac {1}{1.2} + \frac {1}{1.2.3} + \dots )(\rm 1 - \frac {1}{1.2} + \frac {1}{1.2.3} - \dots )$

Sum the following into infinity

$(\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 2^2}{3!} + \dots )$