Taylor Expansion of Arcsin(1/x) and Its Convergence

The Taylor expansion of the Arcsine function presents an interesting challenge when considering arcsin(1/x). Unlike the standard Taylor series for arcsin(x), which is centered at x0, the function arcsin(1/x) is only defined for xge;1. This article will explore the Taylor series expansion for arcsin(1/x) and address the convergence issues that arise.

Introduction to Taylor Series Expansion

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The general form of a Taylor series expansion of a function f(x) around a point xa is given by:

f(x) f(a) f'(a)(x-a) frac{f''(a)}{2!}(x-a)^2 frac{f'''(a)}{3!}(x-a)^3 ldots

Taylor Series of Arcsin(x) at x0

The Taylor series for the arcsin(x) function around x0 is well-known and given by:

arcsin(x) x frac{x^3}{3} frac{3x^5}{10} frac{5x^7}{42} ldots

This series is valid for |x|le;1. However, when we consider the function arcsin(1/x), the domain of definition changes to xge;1.

Expanding the Series for Arcsin(1/x)

Substitution and Simplification

To find the Taylor series expansion for arcsin(1/x), we substitute x with 1/x in the series of arcsin(x):

arcsin(1/x) frac{1}{x} frac{1}{3x^3} frac{3}{1^5} frac{5}{42x^7} ldots

This series is valid for 1/x le; 1, or equivalently, x ge; 1.

Derivatives and Evaluation at x1

For a more precise expansion around x1, we need to re-evaluate the function and its derivatives at x1.

The first derivative of arcsin(1/x) is:

frac{d}{dx} arcsin(1/x) frac{-1}{sqrt{1 - 1/x^2}} cdot frac{-1}{x^2} frac{1}{x^2 sqrt{1 - 1/x^2}}

Evaluating at x1, we encounter a division by zero, indicating that care must be taken with the series expansion, particularly concerning convergence.

Conclusion

The Taylor expansion of arcsin(1/x) depends on the behavior of the function near the point of interest. For practical purposes, if you need an explicit series expansion for arcsin(1/x), you may have to consider a specific form or numerical methods for values of x ge; 1 to ensure convergence and correct evaluation.

For further calculations or a specific form of the series, contact the author for assistance. It is important to note the limitations and care required in the evaluation of such series expansions.