Answer
We prove that $f\left( x \right) = {\left( {1 + x} \right)^{100}}$ is represented by the Maclaurin series $\mathop \sum \limits_{n = 0}^\infty \left( {\begin{array}{*{20}{c}}
{100}\\
n
\end{array}} \right){x^n}$, for $\left| x \right| \lt 1$.
Work Step by Step
From Table 2, we know that the function ${\left( {1 + x} \right)^a}$ is represented by the Maclaurin series $\mathop \sum \limits_{n = 0}^\infty \left( {\begin{array}{*{20}{c}}
a\\
n
\end{array}} \right){x^n}$, for $\left| x \right| \lt 1$, where $a$ is any number (integer or not).
Using this fact, therefore, $f\left( x \right) = {\left( {1 + x} \right)^{100}}$ is represented by the Maclaurin series $\mathop \sum \limits_{n = 0}^\infty \left( {\begin{array}{*{20}{c}}
{100}\\
n
\end{array}} \right){x^n}$, for $\left| x \right| \lt 1$.
