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

True/False

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?