Find the first three nonzero terms of the Maclaurin series for the function by using an appropriate substitution in a known Maclaurin series and performing any algebraic operations that are required.

Question

Find the first three nonzero terms of the Maclaurin series for the function  by using an appropriate substitution in a known Maclaurin series and performing any algebraic operations that are required.

Quizplus · Accepted Answer

We can start with the Maclaurin series for $\frac{1}{1-x}$ which is $1+x+x^2+x^3+...$. If we substitute $-2x$ for $x$, we get the Maclaurin series for $\frac{1}{1+2x}$ which is $1-2x+4x^2-8x^3+...$. Multiplying this by $2$ will give us the first few terms of the Maclaurin series for $e^{4x}$ which is $2-4x+8x^2-...$. Finally, if we substitute $\frac{1}{4}$ for $x$, we get the first few terms of the Maclaurin series for $e$. Thus, the first three nonzero terms of the Maclaurin series for $2e^{4x}$ are $2-8x+32x^2$. This matches choice C.

Find the First Three Nonzero Terms of the Maclaurin Series