I am trying to find the taylor series for (x,y) = (x+y)^n. I began by getting the partial derivatives as df/dx = n(x+y)^(n-1) (1+y) and df/dy=n(x+y)^(n-1) (x+1). don't know where to go from there.

Question

I am trying to find the taylor series for (x,y) = (x+y)^n.  I began by getting the partial derivatives as df/dx = n(x+y)^(n-1) (1+y) and df/dy=n(x+y)^(n-1) (x+1).  don't know where to go from there.

amistre64 · Answer

yeah, the taylor of multiple variables takes partials and attributes then to powers ....

amistre64 · Answer

and it might be easier to keep track of if you notate it as:  Fx, Fy, Fxx, Fyy, Fxy, Fyx, ...

anonymous · Answer

sorry, need a little more guidance as to where to go from here.

amistre64 · Answer

well, from what i recall, the subscripts help to define the variable exponents

Fx is the coeff of x
Fxx is the coeff of x^2
Fxy is the coeff of xy

etc

amistre64 · Answer

in a single variable setup that relates to

Fx is coeff of x
Fxx is coeff of x^2
Fxxx is coeff of x^3
etc ....

amistre64 · Answer

(x+y)^n
$$
F_x = n(\hat x+\hat y)^{n-1}\
F_{x^{(2)}} = n(n-1)(\hat x+\hat y)^{n-2}\
F_{x^{(3)}} = n(n-1)(n-2)(\hat x+\hat y)^{n-3}\
F_{x^{(k)}} = n(n-1)(n-2)...(n-k+1)(\hat x+\hat y)^{n-k}\
$$

the Fys will have the same structure of course

its the intermediarys of xy  x^2y  xy^2 and such that can get harrowing