Problem 56 - Essential Calculus

Question

In Exercises \(55-58,\) the series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !} $$

Step-by-Step Solution

Answer

The function represented by the series is the sine function.

Step 1: Calculate the partial sum

Use a computer algebra system to calculate the partial sum \(S_{10}\) of the series. To do this, replace the upper limit of the sum from \(\infty\) to \(10\). This gives the partial sum \(S_{10} = \sum_{n=0}^{10}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}\).

Step 2: Graph the partial sum

With the use of the computer algebra system, graph \(S_{10}\) on a set of axes. Observe the shape, intersection points, and general behavior of the graph.

Step 3: Identify the function

Compare the graph of the partial sum \(S_{10}\) to the plots of well-known functions. The shape of the graph should resemble one of these functions. After analyzing the graph, it can be determined that the graph matches the graph of sine due to its wave-like characteristics. This leads to the conclusion that the original series represents the sine function.

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry, represented as \( \sin(x) \). It is known for its periodic wave-like behavior, repeating every \( 2\pi \) along the x-axis. This pattern is often associated with circular motion and oscillations such as sound waves or light. The sine function is essential in various fields including physics, engineering, and even finance where cyclical trends can occur.

Some key properties of the sine function include:

• Range: The values of sine function range between -1 and 1.
• Periodicity: Completes a full cycle over an interval of \( 2\pi \).
• Odd Function: Reflects over the origin, meaning \( \sin(-x) = -\sin(x) \).

The Taylor series expansion is a common method to represent the sine function mathematically, especially useful when trying to approximate its value for certain inputs. This is precisely what happens in trigonometric series expansions, where complex functions like sine are expressed as an infinite sum of terms involving powers of \( x \) and factorials in the denominators.

Partial Sums

A partial sum is the sum of the first \( n \) terms of a sequence or a series. When dealing with infinite series, calculating the partial sum helps us approximate the solution with increasing accuracy, as more terms are calculated.

In our exercise, we compute the 10th partial sum \( S_{10} \) of the infinite series that defines the sine function. The task was to compute \( S_{10} = \sum_{n=0}^{10} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \), where:

• \((-1)^n\) accounts for alternating signs, which contributes to the sine function's wave nature.
• \(x^{2n+1}\) corresponds to the power of the variable \( x \).
• \((2n+1)!\) or the factorial of \( (2n+1) \) in the denominator ensures convergence.

Approximating the sine function with partial sums allows mathematicians and engineers to make real-world calculations possible using a limited number of terms. As the number of terms increases, the partial sum \( S_n \) approaches the true value of the sine function more closely.

Computer Algebra System

A computer algebra system (CAS) is a type of software that facilitates symbolic mathematics. Unlike numerical calculators that provide approximate answers, CAS tools can handle exact expressions including variables, algebraic equations, and symbolic integrals or derivatives.

Using a CAS, like Mathematica or MATLAB, assists users in managing complex mathematical operations like computing partial sums and graphing functions based on series expansions.

• Graphing: CAS can generate plots of functions specified by a series or algebraic expression.
• Symbolic Computation: Allows for manipulation of mathematical expressions, solving equations, and performing calculus tools without numerical approximation.
• Complex Calculations: With the aid of CAS, intricate series computations and manipulations are streamlined.

In this context, a CAS was used to calculate and graph the partial sum \( S_{10} \), allowing students and researchers to visually compare the computed graph to known functions. This visual representation helps in identifying that the series represents the sine function accurately.