Understanding the Taylor Series Expansion of arcsin(x)

arc sin(x) is a crucial function in mathematics, particularly in trigonometry. The Taylor series expansion of arc sin(x) can be derived by leveraging its integral form and applying binomial series expansion. This detailed guide will explore this process in depth.

Integral Form of arc sin(x)

The integral form of the inverse sine function is given by:

arc sin(x) ∫ 1/√(1-t2) dt

This integral can be expressed in an infinite series form, reflecting the behavior of the function near the origin. For |x|1, the series expansion is:

arc sin(x) x (1/2)x3/3! (3/4)x5/5! (3middot;5/4middot;6)x7/7! ...

Derivative of arc sin(x)

The derivative of arc sin(x) with respect to x is:

arc sin'(x) 1/√(1-x2)

A binomial series expansion can be applied to this derivative:

1/√(1-x2) 1 - (1/2)x2 (1/2middot;3/2)x4 - (1/2middot;3/2middot;5/2)x6 ...

Integrating the Binomial Series Expansion

By integrating both sides of the binomial series expansion of the derivative, we can derive the Taylor series of arc sin(x):

Integrating the first term:

∫1 dx x

Integrating the second term:

∫- (1/2)x2 dx - (1/2)x3/3 - (1/6)x3

Integrating the third term:

∫(1/2middot;3/2)x4 dx (1/2middot;3/2middot;4/5)x5/5 (3/20)x5

Continuing this process, the resulting Taylor series expansion is:

arc sin(x) ≈ x - (1/6)x3 (3/40)x5

The Significance of the Taylor Series Expansion

The Taylor series expansion of arc sin(x) is significant because it helps approximate the function for values of x close to zero. It also aids in various mathematical and computational applications, such as numeric solutions to differential equations and understanding the behavior of certain functions in complex analysis.

Conclusion

In this detailed guide, we have explored the Taylor series expansion of arc sin(x). By leveraging the integral form of the function and applying the binomial series expansion, we can derive an approximate series that is useful for numerical and theoretical purposes. This expansion is a key tool in the field of mathematics and is essential for understanding and working with inverse trigonometric functions.