Answer
$\frac{1-{{x}^{20}}}{1-{{x}^{2}}}$
Work Step by Step
The provided series,
$1+{{x}^{2}}+{{x}^{4}}+{{x}^{6}}+\cdots $.
Here, ${{a}_{1}}=1$, $n=10$ and
$\begin{align}
& r=\frac{x}{1} \\
& =x
\end{align}$
Substitute ${{a}_{1}}=1$, $n=10$ and $r=x$ in ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}$
$\begin{align}
& {{S}_{10}}=\frac{1\left[ 1-{{\left( {{x}^{2}} \right)}^{10}} \right]}{1-{{x}^{2}}} \\
& =\frac{1-{{x}^{20}}}{1-{{x}^{2}}}
\end{align}$
Thus, the sum of the first ten terms, ${{S}_{10}}$ for $1+{{x}^{2}}+{{x}^{4}}+{{x}^{6}}+\cdots $ is$\frac{1-{{x}^{20}}}{1-{{x}^{2}}}$.
