Understanding Taylor's Expansion of arctan x and arcsin x

Series expansions for mathematical functions, such as arctan(x) and arcsin(x), are fundamental tools in mathematical analysis and computational methods. Consider the Taylor series expansions of these functions, which can be derived either by directly obtaining coefficients using the formula a_n f^{(n)}(0) / n! or by integrating term by term the series for 1/sqrt{1-x^2} and 1/(1 x^2).

Series Expansion for arctan x and arcsin x

The Taylor series expansion for arctan(x) and arcsin(x) are given as follows:

arcsin(x) x - frac{1}{6}x^3 frac{3}{40}x^5 - frac{5}{112}x^7 cdots arctan(x) x - frac{1}{3}x^3 frac{1}{5}x^5 - frac{1}{7}x^7 cdots

For comparison, the Taylor series expansions for sin(x) and tan(x) are:

sin(x) x - frac{1}{6}x^3 frac{1}{120}x^5 - frac{1}{5040}x^7 cdots tan(x) x frac{1}{3}x^3 frac{2}{15}x^5 frac{17}{315}x^7 cdots

Evaluating a Limit Using Taylor Series

Consider the limit

displaystyle lim_{x to 0} frac{arcsin(x) - sin(x)}{tan(x) - arctan(x)}

Let's break this down step by step. First, substitute the Taylor series expansions:

arcsin(x) - sin(x) x^3 left( frac{1}{3} - frac{1}{6} right) x^5 left( frac{3}{40} - frac{1}{120} right) cdots frac{1}{6}x^3 - frac{1}{120}x^5 cdots tan(x) - arctan(x) x^3 left( frac{2}{3} - frac{1}{3} right) x^5 left( -frac{1}{15} frac{1}{15} right) cdots frac{2}{3}x^3 x^5 left( -frac{1}{15} frac{1}{15} right) cdots frac{2}{3}x^3

Thus, the limit simplifies to:

displaystyle lim_{x to 0} frac{x^3 left( frac{1}{6} - frac{1}{120} right) cdots}{frac{2}{3}x^3} lim_{x to 0} frac{frac{1}{6} - frac{1}{120} cdots}{frac{2}{3}} frac{frac{1}{6} - frac{1}{120}}{frac{2}{3}} frac{frac{20 - 1}{120}}{frac{2}{3}} frac{19}{120} cdot frac{3}{2} frac{19}{80}

However, we need to check the exact coefficient of x^3 in the numerator and find the correct limit. Using the exact coefficients, the terms simplify to:

displaystyle lim_{x to 0} frac{frac{1}{6}x^3 cdots}{frac{2}{3}x^3} frac{frac{1}{6} - frac{1}{120}}{frac{2}{3}} frac{frac{20 - 1}{120}}{frac{2}{3}} frac{19}{80} cdot frac{3}{2} frac{1}{2}

Hence, the correct limit is displaystyle lim_{x to 0} frac{arcsin(x) - sin(x)}{tan(x) - arctan(x)} frac{1}{2}.

Visualizing the Functions Using Scilab

Now, let's analyze the behavior of the functions around (x 0) using Scilab. We will write the code to plot these functions and observe their behavior as (x) approaches zero.

poly 'x'; x -0.1:0.01:0.1; figure(figsize (10, 10)) subplot(2, 2, 1) publish_plot(arcsin(x), 'arcsin(x)', 'x', 'arcsin(x)', 'title', 'arc Sin(x) near 0') subplot(2, 2, 2) publish_plot(sin(x), 'sin(x)', 'x', 'sin(x)', 'title', 'sin(x) near 0') subplot(2, 2, 3) publish_plot(tan(x), 'tan(x)', 'x', 'tan(x)', 'title', 'tan(x) near 0') subplot(2, 2, 4) publish_plot(arctan(x), 'arctan(x)', 'x', 'arctan(x)', 'title', 'arctan(x) near 0')

By plotting these functions, you can observe how each function behaves around (x 0).

Approximating with Polynomials

For angles very close to (0), the functions sin(x) and tan(x) can often be approximated by (x), as their derivatives are (1) at (x 0). Correspondingly, their inverses arcsin(x) and arctan(x) are also approximated by (x). Hence, we can conclude:

arcsin(x) - sin(x) approx 0 and tan(x) - arctan(x) approx 0

Thus, the limit simplifies to:

displaystyle lim_{x to 0} frac{arcsin(x) - sin(x)}{tan(x) - arctan(x)} 1