#### Answer

$\displaystyle \frac{3}{8}$

#### Work Step by Step

The sum has five terms, for i=0,1,2,3,4,5.
with alternating signs, of which the first is a plus, $(-1)^{0}=+1.$
The denominators progress as $0!,1!,2!, 3!,4!$
that is as 1, 1, 2, 6, 24.
Sum = $+\displaystyle \frac{1}{1}-\frac{1}{1}+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}$
$=\displaystyle \frac{1}{2}-\frac{1}{6}+\frac{1}{24}=\frac{12-4+1}{24}=\frac{9}{24}=\frac{3}{8}$