The first four terms in the power series expansion of the function $f ( x ) = e ^ { 2 x }$ about $x = 0$ are
A) $1 + x + x ^ { 2 } + x ^ { 3 }$
B) $1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 }$
C) $1 + 2 x + 2 x ^ { 2 } + 4 x ^ { 3 } / 3$
D) $1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 } / 3$
E) $1 + 2 x + 4 x ^ { 2 } + 8 x ^ { 3 }$

Question

Quizplus · Accepted Answer

The power series expansion of $e^{2x}$ is given by $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$. The first four terms are $1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} = 1 + 2x + 2x^2 + \frac{4x^3}{3}$.

The First Four Terms in the Power Series Expansion of the Function