Write the Maclaurin series for tan⁻¹ x.

The Maclaurin series is a powerful tool in calculus that allows us to express functions as infinite sums of their derivatives at a single point, specifically at zero. For the function \( \tan^{-1}(x) \), we can derive its Maclaurin series by finding its derivatives and evaluating them at \( x = 0 \). Let’s walk through the process step by step.

Understanding the Function

The function \( \tan^{-1}(x) \), also known as the inverse tangent function, is defined for all real numbers. Its behavior is particularly interesting because it approaches \( \frac{\pi}{2} \) as \( x \) approaches infinity and \( -\frac{\pi}{2} \) as \( x \) approaches negative infinity. However, for our purposes, we will focus on its expansion around \( x = 0 \).

Finding the Derivatives

To construct the Maclaurin series, we need to calculate the derivatives of \( \tan^{-1}(x) \) at \( x = 0 \). The first few derivatives are:

• First derivative: \( f'(x) = \frac{1}{1+x^2} \)
• Second derivative: \( f''(x) = -\frac{2x}{(1+x^2)^2} \)
• Third derivative: \( f'''(x) = \frac{2(3x^2-1)}{(1+x^2)^3} \)
• Fourth derivative: \( f^{(4)}(x) = \frac{24x}{(1+x^2)^4} \)

Now, we evaluate these derivatives at \( x = 0 \):

• \( f(0) = \tan^{-1}(0) = 0 \)
• \( f'(0) = 1 \)
• \( f''(0) = 0 \)
• \( f'''(0) = -2 \)
• \( f^{(4)}(0) = 0 \)

Constructing the Series

The Maclaurin series is given by the formula:

\( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4 + \ldots \)

Substituting the values we found:

• \( f(0) = 0 \)
• \( f'(0)x = 1x \)
• \( \frac{f''(0)}{2!}x^2 = 0 \)
• \( \frac{f'''(0)}{3!}x^3 = -\frac{2}{6}x^3 = -\frac{1}{3}x^3 \)
• \( \frac{f^{(4)}(0)}{4!}x^4 = 0 \)

Thus, the series up to the \( x^3 \) term is:

\( \tan^{-1}(x) \approx x - \frac{1}{3}x^3 + \ldots \)

General Term of the Series

To find a more general expression, we can observe a pattern in the derivatives. The series can be expressed as:

\( \tan^{-1}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \)

This series converges for \( |x| \leq 1 \) and provides a way to compute \( \tan^{-1}(x) \) for values within this range.

Practical Applications

The Maclaurin series for \( \tan^{-1}(x) \) is not just a theoretical exercise; it has practical applications in numerical methods, physics, and engineering, particularly in situations where you need to approximate the inverse tangent function quickly and efficiently.

In summary, the Maclaurin series for \( \tan^{-1}(x) \) is a useful representation that allows us to understand and compute the function near zero, showcasing the beauty of calculus in approximating complex functions with simple polynomial expressions.