Approximating e Using Taylor Expansion: A Comprehensive Guide

Understanding how to approximate functions using Taylor series is a fundamental skill in mathematical analysis and has numerous practical applications. One such function is ex, which is of particular interest due to its unique properties and ubiquity in various fields of science and engineering. In this article, we will delve into how to use Taylor expansion to approximate the value of ex at a specific point, specifically when x 1. We will also discuss the Lagrange form of the reminder and how it helps ensure the accuracy of our approximation.

The Taylor Series for ex

The Taylor series of a function f(x) around a point x_0 is given by:

fx sum_{k0}^n frac{f^{k}x_0}{k!}(x-x_0^k) R_n

where R_n is the Lagrange form of the remainder, which is given by:

R_n frac{f^{n 1} xi}{(n 1)!}(x - x_0)^{(n 1)}

xi is a value between x_0 and x for which R_n is evaluated.

Approximating e1

Let's consider the specific case where x_0 0 and x 1. The Taylor series of ex around x_0 0 is:

exp x sum_{k0}^n frac{1}{k!} x^k R_n

Substituting x 1 and x_0 0, we get:

exp 1 1 1/1! 1/2! 1/3! ... R_n

The Lagrange form of the remainder for ex is:

R_n frac{e^{xi}}{(n 1)!}

where xi is some value between 0 and 1.

Choosing the Number of Terms

To ensure that our approximation is accurate to within a certain tolerance, we need to choose a sufficient number of terms in the series. In our case, we want to ensure that R_n is less than 1/10. Therefore, we need to find the smallest n such that:

frac{e^{xi}}{(n 1)!}

Since xi is between 0 and 1, the maximum value of e^{xi} is 3 (as xi 1).

Calculating the Required Terms

To find the smallest n such that frac{3}{(n 1)!} , we can solve the inequality:

Taking the factorial of 30:

30! 2.6525285981219105864e 32

We can see that:

This implies:

n 1 > 30

n > 29

Therefore, we need to include terms up to the 5th power of x to ensure our approximation is accurate to within 1/10. This means:

n 4

So, the Taylor series up to the 4th term is:

exp 1 1 1 1/2 1/6 1/24

Simplifying this, we get:

exp 1 approx 2.71875

Which is a reasonably accurate approximation of e 2.71828.

Conclusion

Using Taylor expansion, we can approximate the value of ex to a high degree of accuracy. The Lagrange form of the remainder helps us ensure that our approximation is within a desired tolerance. In this case, we used the series up to the 4th term to approximate e1 as 2.71875, which is a close approximation to the true value of e. This method can be applied to approximate other functions and is a powerful tool in mathematical analysis.