Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{4 x}{(1-x)^{2}}$ .
A) $4+4 x^{3}+4 x^{5}+4 x^{7}+\cdots$
B) $4+4 x^{2}+4 x^{3}+4 x^{4}+\cdots$
C) $4 x+8 x^{2}+12 x^{3}+16 x^{4}+\cdots$
D) $4+8 x+12 x^{2}+16 x^{3}+\cdots$

Question

Quizplus · Accepted Answer

The Taylor series for $\frac{1}{1-x}$ about 0 is $1 + x + x^2 + x^3 + \cdots$. Differentiating this gives $1 + 2x + 3x^2 + 4x^3 + \cdots$. Multiplying by $4x$ gives the series for $\frac{4x}{(1-x)^2}$, which is $4x + 8x^2 + 12x^3 + 16x^4 + \cdots$.

Use the Derivative of the Taylor Series About 0 For \frac{1}{1-x}