Problem 24 - Calculus: Early Transcendentals

Question

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \tan^{-1} (2x) \)

Step-by-Step Solution

Answer

As \( n \) increases, the partial sums better approximate \( f(x) = \tan^{-1}(2x) \).

Step 1: Identify the Function to Expand

We need to find the power series representation for the function \( f(x) = \tan^{-1}(2x) \). The inverse tangent function \( \tan^{-1}(x) \) can be expressed as a power series.

Step 2: Use the Known Series for \( \tan^{-1}(x) \)

The power series for \( \tan^{-1}(x) \) is given by:\[\tan^{-1}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}.\] This series is derived from the geometric series formula and integration.

Step 3: Substitute \( 2x \) into the Series

To find the series for \( \tan^{-1}(2x) \), substitute \( 2x \) for \( x \) in the series:\[\tan^{-1}(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} (2x)^{2n+1}.\] This adjusts the series to represent \( f(x) = \tan^{-1}(2x) \).

Step 4: Simplify the Series Terms

Simplify the expression for each term in the series:\[\tan^{-1}(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n+1}}{2n+1} x^{2n+1}.\] These are the terms of the power series expansion for \( \tan^{-1}(2x) \).

Step 5: Graph the Function and Partial Sums

Graph the function \( f(x) = \tan^{-1}(2x) \) and several partial sums \( s_n(x) = \sum_{k=0}^{n} \frac{(-1)^k 2^{2k+1}}{2k+1} x^{2k+1} \) on the same axes. Observe how as \( n \) increases, \( s_n(x) \) approximates \( f(x) \) more closely.

Step 6: Observe Convergence as \( n \) Increases

As \( n \) increases, the partial sums \( s_n(x) \) converge to the actual function \( \tan^{-1}(2x) \) over its interval of convergence. Larger \( n \) results in better approximation outside the immediate vicinity of \( x = 0 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the usual trigonometric functions, such as sine, cosine, and tangent. These functions include \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \), among others.
Inverse tangent, denoted as \( \tan^{-1}(x) \), gives the angle whose tangent is \( x \). It is important in applications involving angles and modeling in trigonometry.

• Range of \( \tan^{-1}(x) \): Typically \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
• Understanding inverse trigonometric functions is crucial for solving equations where you need to find angles based on a given tangent value.
• \( \tan^{-1}(2x) \) implies we are finding the angle whose tangent is \( 2x \).

The power series expansion simplifies complex functions into an infinite sum of terms, making inverse trigonometric functions easier to work with in calculations.
This transformation is achieved through mathematical series, allowing for approximations and easy integration or differentiation where applicable.

Convergence of Series

Convergence of a series refers to whether the infinite sum of its terms approaches a finite limit as more terms are added. In dealing with power series for inverse trigonometric functions, determining convergence is key to ensuring accurate representations and calculations.

• The series for \( \tan^{-1}(x) \), expressed as \( \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1} \), converges for \( |x| \leq 1 \), making it valid for many practical applications.
• By substituting \( 2x \) for \( x \), the convergence still needs to be checked over the new domain \(-0.5 \leq x \leq 0.5 \).
• As \( n \) increases, the partial sums \( s_n(x) \) approximate \( \tan^{-1}(2x) \) more closely across the interval, demonstrating convergence.

Understanding convergence helps avoid errors in computation by ensuring the series fits the desired interval. This is especially useful as computations become more complex or as you're working with boundary values in series.

Graphical Representation of Functions

A graphical representation of functions provides a visual way to understand how a function behaves over its domain. For inverse trigonometric functions like \( f(x) = \tan^{-1}(2x) \), this visual element is especially useful in interpreting the function's characteristics.

• Graphing \( \tan^{-1}(2x) \) alongside its power series approximations helps see the accuracy and convergence of the series.
• As more terms are added (increasing \( n \)), the power series representation \( s_n(x) \) hugs the actual curve more closely, demonstrating convergence.
• This approach also allows the observation of how the function approaches its asymptotes, thanks to the finite nature of its range.

This visual insight is also beneficial in checking the validity of the series approximation over different intervals and recognizing the limits of approximation overtly when considering finite sections vs. the entire function.