# LaGrange Error and power series

• SoaringQuail
In summary, the conversation discusses finding a formula for the truncation error when using a specific approximation to approximate 1/(1-x^2) over a given interval. The use of LaGrange error is mentioned and it is suggested to use the formula where M is an upper bound on the n+1 derivative. The value of n+1 can either be 7 or 4 depending on the variable used in the formula. Further guidance is requested on how to proceed with the problem.

#### SoaringQuail

There's a homework problem that I've been struggling over:

Find a formula for the truncation error if we use 1 + x^2 + x^4 +x^6 to approximate 1/(1-x^2) over the interval (-1, 1).

Now, I assume that you need to use LaGrange error but I'm not sure how to proceed. Any help would be greatly appreciated.

SoaringQuail said:
There's a homework problem that I've been struggling over:

Find a formula for the truncation error if we use 1 + x^2 + x^4 +x^6 to approximate 1/(1-x^2) over the interval (-1, 1).

Now, I assume that you need to use LaGrange error but I'm not sure how to proceed. Any help would be greatly appreciated.

Since you mention LaGrange error, presumably you know the formula for it! Off the top of my head, I believe it is
$$E\le \frac{M}{(n+1)!}|x-a|^{n+1}$$
where M is an upper bound on the n+ 1 derivative. Here, by the way, you can take either n+1= 7 or replace "x2" by "y" and use n+1= 4 with the formula in y.

## 1. What is the LaGrange error in power series?

The LaGrange error in power series is a mathematical concept used to estimate the error in approximating a function using a finite number of terms in its Taylor series. It is named after the mathematician Joseph-Louis LaGrange and is also known as the remainder term.

## 2. How is the LaGrange error formula derived?

The LaGrange error formula is derived using Taylor's theorem, which states that any function can be approximated by its Taylor polynomial with a remainder term that depends on the value of the function's derivative at a certain point. By using the remainder term, we can estimate the error in the approximation of the function.

## 3. What is the significance of the LaGrange error in power series?

The LaGrange error allows us to determine the accuracy of a Taylor series approximation and to improve the approximation by adding more terms. It is also useful in determining the convergence of a power series, which is important in various fields such as physics, engineering, and economics.

## 4. How is the LaGrange error used in real-world applications?

The LaGrange error is used in various real-world applications where approximating functions is necessary. For example, in physics, it is used to estimate the error in approximating the motion of a particle using a finite number of terms in its Taylor series. In finance, it is used to calculate the error in approximating the value of an investment using a Taylor series.

## 5. Can the LaGrange error be negative?

Yes, the LaGrange error can be negative. This indicates that the Taylor polynomial overestimates the value of the function at a certain point. However, as we add more terms to the polynomial, the error decreases and eventually becomes positive, indicating that the polynomial is a better approximation of the function at that point.