Answer
The simplified solution is $\frac{1}{x}-\frac{1}{x+2}$ and the sum is $\frac{100}{101}$.
Work Step by Step
Let us consider the rational expression.
$\begin{align}
& \frac{2}{x\left( x+2 \right)}=\frac{A}{x}+\frac{B}{x+2} \\
& =\frac{A\left( x+2 \right)+Bx}{x\left( x+2 \right)}
\end{align}$
$2=A\left( x+2 \right)+Bx$ (I)
Substitute the value of $x=-2$ in the equation (I),
$\begin{align}
& 2=B\left( -2 \right) \\
& B=-1
\end{align}$
Again, putting the value of $x=0$ in (I),
$\begin{align}
& 2=A\left( 2 \right) \\
& A=1
\end{align}$
Therefore, $\frac{2}{x\left( x+2 \right)}=\frac{1}{x}-\frac{1}{x+2}$ is the simplified form of the result.
And use the above result to find the sum of the series,
$\begin{align}
& \frac{2}{1\cdot 3}+\frac{2}{3\cdot 5}+\frac{2}{5\cdot 7}+\cdots +\frac{2}{99\cdot 101}=\left( \frac{1}{1}-\frac{1}{3} \right)+\left( \frac{1}{3}-\frac{1}{5} \right)+\left( \frac{1}{5}-\frac{1}{7} \right)+\cdots +\left( \frac{1}{99}-\frac{1}{101} \right) \\
& =\frac{1}{1}-\frac{1}{101} \\
& =\frac{100}{101}
\end{align}$
Thus, the sum is $\frac{100}{101}$.
