Find the Taylor series expansion for f(x) = $\frac { x } { 1 - x }$ and use it to determine which of the following is false?
A) $\frac { x } { 1 - x }$ = x + $x ^ { 2 }$ + $x ^ { 3 }$ + $x ^ { 4 }$ +...
B) $\frac { 1 } { 2 }$ = $\frac { 1 } { 3 }$ + $\left( \frac { 1 } { 3 } \right) ^ { 2 }$ + $\left( \frac { 1 } { 3 } \right) ^ { 3 }$ + ...
C) - $\frac { 1 } { 3 }$ = $\left( - \frac { 1 } { 2 } \right)$ + $\left( - \frac { 1 } { 2 } \right) ^ { 2 }$ + $\left( - \frac { 1 } { 2 } \right) ^ { 3 }$ + ...
D) - $\frac { 3 } { 5 }$ = $\left( \frac { 3 } { 2 } \right)$ + $\left( \frac { 3 } { 2 } \right) ^ { 2 }$ + $\left( \frac { 3 } { 2 } \right) ^ { 3 }$ +...
E) All the statements are true.

Question

Quizplus · Accepted Answer

The Taylor series expansion for $f(x) = \frac{x}{1-x}$ is $x + x^2 + x^3 + x^4 + \ldots$, which converges for $|x| < 1$. Choices A, B, and C correctly apply this formula, but choice D incorrectly applies it to $x = \frac{3}{2}$, which is outside the radius of convergence.

Find the Taylor Series Expansion for F(x) = x1−x\frac { x } { 1 - x }1−xx​

Find the Taylor Series Expansion for F(x) = $\frac { x } { 1 - x }$