Taylor series of two variable ?

[rclakmal]

Homework Statement

I want to know that how to calculate the required number of terms to obtain a given decimal accuracy in two variable Taylor series .
In one variable case i know there is an error term R(n)=[ f(e)^(n+1)* (x-c)^(n+1)] / (n+1)! where 'e' is between x and c (c is the center value ) so if we want a 3 decimal place accuracy we can have the inequality below

R(n)<=5*10^(-4) then we can obtain particular ' n ' to have that accuracy

But i don't know how to use this in two variable case .please give some explanation or a good tutorial where i can learn it .

 

[HallsofIvy]

The corresponding formula for two variables is
$$R(n)= \sum_{i=0}^{n+1} \frac{\frac{\partial^{n+1} f(e)}{\partial x^{i}\partial y^{n+1-i}}(x-c)^{n+1}}{(n+1)!}$$

 

[Architeuthis Dux]

I can suggest a few things that may help you in calculating the required number of terms for a given decimal accuracy in a two variable Taylor series.

Firstly, it is important to understand the concept of a Taylor series in two variables. A Taylor series in two variables is an expansion of a function in terms of the variables x and y, around a given point (c, d). The general form of a two variable Taylor series is:

f(x,y) = f(c,d) + (x-c)f_x(c,d) + (y-d)f_y(c,d) + (x-c)^2 f_xx(c,d) + (y-d)^2 f_yy(c,d) + (x-c)(y-d)f_xy(c,d) + ...

where f_x, f_y, f_xx, f_yy, f_xy, etc. are the partial derivatives of the function f with respect to x and y evaluated at the point (c,d).

To calculate the required number of terms for a given decimal accuracy, you can use the same error term formula as in the one variable case, but with a slight modification. The error term in two variables can be written as:

R(n) = [ f(e1,e2)^(n+1) * (x-c)^(n+1) * (y-d)^(n+1) ] / (n+1)!

where e1 and e2 are values between x and c, and y and d, respectively.

To obtain a 3 decimal place accuracy, you can set the error term to be less than or equal to 5*10^(-4). This will give you an inequality that includes both x and y variables. Solving this inequality may be challenging, but you can use numerical methods or computer software to find the required number of terms.

Another useful approach is to use the Lagrange form of the remainder, which gives a more accurate estimation of the error term. This can be helpful in cases where the error term from the above formula is too large.

I would also recommend referring to textbooks or online resources specifically on two variable Taylor series for a more detailed explanation and examples. Good luck with your studies!