I was given the function: $f(x) = 1/(1+x)^2$ and its Taylor series: $1 - 2x + 3x^2 - 4x^3 + \cdots$
In order to get the Taylor series for the closely related function $1/((1/2)+x)^2$, I simply substituted $(x-(1/2))$ for $x$ in the Taylor series given to me:
$$1-2(x-(1/2)) + 3(x-(1/2))^2 - 4(x-(1/2))^3 +\cdots$$
This seems straight forward enough, but xMaxima gives me a different answer for the Taylor series of $f(x)$, it gives this:
$$4-16\cdot x+48\cdot x^2-128\cdot x^3+320\cdot x^4+\cdots$$
I also get this answer if I represent $f(x)$ as $1/(1/2)^2 \cdot 1/(1+2x)$ and then substitute the $x$'s for $2x$'s in the Taylor series before multiplying each term by $1/(1/2)^2$
What's going on here...Why is the first simpler approach giving me something different, and probably incorrect..