Finding the Taylor Series of f(x) = x/(2+x)

[chemnoob.]

Homework Statement

Obtain the Taylor series in powers of x + 1 for f(x) = x/(2 + x), giving
the general term.

Homework Equations

The Attempt at a Solution

Wrote it out as x*(1/1-(-(x+1)).

 

[hunt_mat]

If
$$f(x)=\frac{x}{x+2}$$
We are asked to write it as a series in $$y=x+1$$, so in terms of y, the function becomes:
$$F(y)=\frac{y-1}{y+1}$$
Now use all your previous knowledge about Taylor series to find the expansion in terms of y

 

[chemnoob.]

hmm.. confused

 

[hunt_mat]

OK, write your function as the following:
$$f(x)=\frac{x}{x+2}=\frac{(x+1)-1}{(x+1)+1}$$
Use all the previous knowledge you have to find the taylor series.

 

[hunt_mat]

Or you could write:
$$f(x)=\frac{x}{x+2}=\frac{x+2-2}{x+2}=1-\frac{2}{x+2}=1-\frac{2}{(x+1)+1}$$
If that makes it easier.