Question

Find the required Fourier series for the given function and sketch the graph of the function to which the series converges over three periods. $$ f(x)=1, \quad 0

Short answer

Question: Find the Fourier sine series for the function \(f(x) = 1\) for \(0 < x < \pi\) and sketch its graph over three periods. Answer: The Fourier sine series for the given function is given as: $$ f(x) \approx \sum_{n=1}^{\infty} \frac{2}{\pi} \frac{1 - (-1)^n}{n} \sin(nx) $$ To sketch the graph, we can plot the sum of the series as we increase the number of terms over the interval \(0 < x < \pi\). The graph will converge to \(f(x)=1\) as more terms are included, with sine wave-like oscillations around the actual function that decrease in amplitude.

Step-by-step solution

Determine sine series formula for the given function

The general formula for the Fourier sine series of a function f(x) with the period of \(2L\) is given as: $$ f(x) \approx \sum_{n=1}^{\infty} b_n \sin \left(\frac{n \pi x}{L}\right) $$ with $$ b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin \left(\frac{n \pi x}{L}\right) dx $$ Since our function has a period of \(2\pi\), we can set \(L = \pi\) and find the coefficients \(b_n\) to determine the sine series.

Find the Fourier coefficients \(b_n\)

Now, let's find the coefficients \(b_n\). We can use the formula mentioned above and plug in the values for the given function: $$ b_n = \frac{2}{\pi} \int_{0}^{\pi} 1 \cdot \sin(nx) dx $$ We can now integrate the sine function: $$ b_n = \frac{2}{\pi} \left[-\frac{1}{n} \cos(nx)\right]_0^\pi = \frac{2}{\pi} \left(-\frac{1}{n} \cos(n\pi) + \frac{1}{n}\right) $$ Since \(\cos(n\pi)=(-1)^n\), we get $$ b_n = \frac{2}{\pi} \left(-\frac{1}{n} (-1)^n + \frac{1}{n}\right) = \frac{2}{\pi} \frac{1 - (-1)^n}{n} $$

Write Fourier sine series for the given function

We can now write the Fourier sine series for the given function using the calculated coefficients \(b_n\): $$ f(x) \approx \sum_{n=1}^{\infty} \frac{2}{\pi} \frac{1 - (-1)^n}{n} \sin(nx) $$ This is the required Fourier sine series for the given function.

Sketch the graph of the function with the determined series

To sketch the graph of the function, we can consider several terms of the series and plot them in a graph. The graph will show the convergence of the series to the actual function over three periods. Since the Fourier sine series converges to the given function only for \(0 < x < \pi\), we'll sketch the graph in this interval. You can use a graphing calculator or software to plot the sum of the series as you increase the number of terms and observe the graph converging to \(f(x)=1\). The graph of the Fourier sine series will have sine wave-like oscillations around the actual function, which will decrease in amplitude as more terms are included in the sum. By analyzing and sketching the graph, we successfully found the required Fourier sine series for the given function and demonstrated its convergence to the original function over three periods.

Key concepts

Fourier sine series

The Fourier sine series is a way to represent a function, particularly a periodic function, using only sine terms. This is especially useful when dealing with functions defined on a specific interval, like from 0 to \( \pi \). Here’s why the sine series can be beneficial:

• Sine functions are naturally periodic, making them ideal for representing periodic features of functions.
• The Fourier sine series only includes terms that involve \( \sin(n\pi x/L) \), which simplifies calculations and avoids the use of cosine terms, especially desirable when working with purely odd functions on a half-range interval.
• This representation helps in approximating complex waveforms using simple sine components, which is critically useful in engineering and physics.

To construct a Fourier sine series, we identify the coefficients \( b_n \), calculated using definite integrals of the function multiplied by the sine function over the given interval.
For a function like \( f(x) = 1 \) in the given problem, the series becomes:\[f(x) \approx \sum_{n=1}^{\infty} b_n \sin(nx)\]This transforms the sometimes complex behavior of functions into a manageable expression built from simple sine waves.

periodic function

A periodic function is one that repeats its values in regular intervals or periods. For mathematical clarity:

• A function \( f(x) \) is periodic with period \( T \) if \( f(x) = f(x + T) \) for all \( x \).
• Periodic functions beautifully model cycles and repeating patterns, like cycles of nature or signals in electronics, thanks to their repetitive characteristics.
• In the context of Fourier series, understanding periodicity is crucial because it lets us model infinite oscillations succinctly and predictably.

The function given in the problem, \( f(x) = 1 \), is defined on \( 0 < x < \pi \) but is considered as periodic by repeating this constant pattern indefinitely over \( 2\pi \).
This implies that if you were to graph it, the same value \( 1 \) will be repeated for each interval of length \( 2\pi \). It's this repeating property that allows us to use Fourier series to create representations that help us understand and utilize the function mathematically and visually.

definite integrals

Definite integrals are used to calculate the accumulation of quantities, often representing areas under curves on a graph. Within the context of Fourier series:

• They provide the means to find coefficients \( b_n \) when decomposing a function into its sine series components.
• In the step-by-step solution, definite integrals are employed to solve \( b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(nx) \, dx \), where the integral calculates the "weight" of each sine term in approximating \( f(x) \).
• This process of integration is critical for ensuring the sine series closely mirrors the original function as the number of terms in the series increases.

By integrating the product of \( f(x) \) and \( \sin(nx) \), you determine how each sine wave (with frequency \( n \)) contributes to the overall series.
Ultimately, these calculations demonstrate how mathematical tools like integrals can be applied to break down complex periodic behaviors into their sine wave constituents and thereby help in reconstructing the function over its defined intervals.