Answer
$-\dfrac{47}{120}$
Work Step by Step
Given that $
a_n=\dfrac{(-1)^n}{2n}
$ then the first 5 terms are
\begin{array}{l}
a_1=\dfrac{(-1)^1}{2(1)}\\\\
a_1=\dfrac{-1}{2}\\\\
a_1=-\dfrac{1}{2}
,\\\\
a_2=\dfrac{(-1)^2}{2(2)}\\\\
a_2=\dfrac{1}{4}
,\\\\
a_3=\dfrac{(-1)^3}{2(3)}\\\\
a_3=\dfrac{-1}{6}\\\\
a_3=-\dfrac{1}{6}
,\\\\
a_4=\dfrac{(-1)^4}{2(4)}\\\\
a_4=\dfrac{1}{8}
,\\\\
a_5=\dfrac{(-1)^5}{2(5)}\\\\
a_5=\dfrac{-1}{10}\\\\
a_5=-\dfrac{1}{10}
.\end{array}
Hence, the sum is $
-\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{8}-\dfrac{1}{10}
=
-\dfrac{47}{120}
.$