Finding a taylor series by substitution

[Ibraheem]

Hello,

In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result.
For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of 1/x, at different centers, to substitute (1+(x^2)/6) for x in 1/x to find the taylor series of 1/(1+(x^2)/6) at a specific center such as a=0. So if we want to find the taylor series of 1/(1+(x^2)/6) centered at 0, can we substitute (1+(x^2)/6) for x in the taylor series of 1/x centered at 1 and then substitute (1+(x^2)/6) for x in the taylor series of 1/x centered at 2 and still get the same result?

 

[Mark44]

Hello,

In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result.
For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of 1/x, at different centers, to substitute (1+(x^2)/6) for x in 1/x to find the taylor series of 1/(1+(x^2)/6) at a specific center such as a=0. So if we want to find the taylor series of 1/(1+(x^2)/6) centered at 0, can we substitute (1+(x^2)/6) for x in the taylor series of 1/x centered at 1 and then substitute (1+(x^2)/6) for x in the taylor series of 1/x centered at 2 and still get the same result?

I don't believe it is.
In any case, it's much simpler to use the Taylor series for a more closely related function, such as f(x) = $\frac{1}{1 + x} = 1 - x + x^2 - x^3 \dots$, and then do the substitution.