Find the Taylor series of f(x) = $\frac { 1 } { 1 + x }$ for $| x |$

Question

Find the Taylor series of f(x) = $\frac { 1 } { 1 + x }$ for $| x |$ < 1 then multiply the series by x to obtain a series expansion of $\frac { x } { x + 1 }$ , and then use these two series to obtain a series expansion of $\frac { 1 - x } { 1 + x }$ for $| x |$ < 1.
Is $\frac { 1 - x } { 1 + x } = 1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 } + 2 x ^ { 4 } + \ldots$ the correct expansion?

Quizplus · Accepted Answer

The Taylor series of $f(x) = \frac{1}{1+x}$ around $x=0$ is $1 - x + x^2 - x^3 + x^4 - \ldots$ for $|x| < 1$. Multiplying this series by $x$ gives the series expansion of $\frac{x}{1+x} = x - x^2 + x^3 - x^4 + \ldots$. To obtain the series expansion of $\frac{1-x}{1+x}$, we subtract the series of $\frac{x}{1+x}$ from the series of $\frac{1}{1+x}$, which results in $1 - 2x + 2x^2 - 2x^3 + 2x^4 - \ldots$, not $1 + 2x + 2x^2 + 2x^3 + 2x^4 + \ldots$.

Find the Taylor Series of F(x) = 11+x\frac { 1 } { 1 + x }1+x1​

Find the Taylor Series of F(x) = $\frac { 1 } { 1 + x }$