Taylor Series for the Absolute Value of sin(x) at Zero

In mathematical analysis, the Taylor series is a powerful tool for approximating functions using polynomials. However, applying the Taylor series to certain functions, such as the absolute value of sin(x) at zero, can lead to interesting challenges and unexpected results. This article will explore the Taylor series for the absolute value of sin(x) centered at zero and delve into why these series may not always exist at certain points.

Introduction to Taylor Series

A Taylor series of a function f(x) centered at x 0 (also known as a Maclaurin series) is an expansion of the function as a sum of terms involving the function's derivatives at that point:

Maclaurin Series Formula

Maclaurin series of sin(x) is given by:

sin(x) x - frac{x^3}{3!} frac{x^5}{5!} - frac{x^7}{7!} cdots sum_{n0}^{infty} frac{-1^n x^{2n1}}{(2n1)!}

Sin(x) and Its Absolute Value

The function sin(x) is an odd function, which means sin(-x) -sin(x). When we consider the absolute value of sin(x), |sin(x)|, the function behaves differently depending on the sign of sin(x). Specifically, |sin(x)| sin(x) for x in [0, π] and |sin(x)| -sin(x) for x in [-π, 0].

Behavior of sin(x)

For small values of x, sin(x) is positive. Therefore, the Taylor series for sin(x) centered at x 0 is valid for small x:

sin(x) x - frac{x^3}{6} frac{x^5}{120} - frac{x^7}{5040} cdots

Thus, the Taylor series for sin(x) at x 0 is:

sin(x) sum_{n0}^{infty} frac{-1^n x^{2n1}}{(2n1)!}

Challenges with Differentiability

The function sin(x) is not differentiable at the origin, x 0. This poses a significant challenge when attempting to find the Taylor series at this point. The function sin(x) is positive for x in (0, π/2) and negative for x in (-π/2, 0), which means the derivative of |sin(x)| is not well-defined at x 0.

Non-Existence of Taylor Series

Given that the function sin(x) is not even continuous at x 0, the Taylor series of |sin(x)| at x 0 cannot exist. This is because the derivative of the function does not exist at this point. This is a fundamental property of functions, particularly those that are not smooth or differentiable at certain points.

Workarounds and Fourier Series

While the Taylor series for |sin(x)| does not exist at x 0, there are alternative methods to approximate the function in this region:

Taylor Series at Other Points

It is possible to find the Taylor series for |sin(x)| at other points where the function is differentiable. For example, centered at x π/2, the 5th order Taylor polynomial expansion is:

frac{1}{24}x^4 - frac{pi}{12}x^3 frac{pi^2 - 8}{16}x^2 - frac{pi^2 - 24}{48}x frac{pi^4 - 48pi^2 384}{384}

This polynomial gives a good approximation on the interval (0, π) with a remainder of less than 0.5.

Fourier Series

An alternative approach is to compute a Fourier series around x 0 over the interval (-π/2, π/2). The 1st order Fourier expansion is:

frac{2}{pi} - frac{4cos{2x}}{3pi}

This approach provides a simpler approximation of the function but may not converge as quickly as a higher-order expansion.

Conclusion

The Taylor series for the absolute value of sin(x) at x 0 does not exist due to the non-differentiability of the function at this point. However, alternative methods such as Taylor series at other points or Fourier series can be used to approximate the function effectively. These methods offer valuable insights into the behavior of functions and their approximations in various scenarios.