Question

This relation between \(\pi\) and the odd positive integers was discovered by Leibniz in 1674 . From the Fourier series for the triangular wave (Example 1 of Section 10.2 ), show that $$ \frac{\pi^{2}}{8}=1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\cdots=\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}} $$

Short answer

Answer: The relation between π and odd positive integers in the form of an infinite sum using the Fourier series for a triangular wave is given by: $$ \frac{\pi^2}{8} = \sum^{\infty}_{n=0} \frac{1}{(2n+1)^{2}} $$

Step-by-step solution

Recall the Fourier series for the Triangular wave

The Fourier series for the triangular wave of period \(2\pi\) can be written as: $$ f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2} \sin(n x) $$ Where \(f(x)\) represents the triangular wave, and the coefficients are given by the expression \(\frac{(-1)^{n-1}}{n^2}\).

Find the value of the Fourier coefficients#\(a_{n}\) and \(b_{n}\)

We know that the triangular wave is an odd function, and the Fourier series of an odd function has only sine terms i.e., for an odd function $$ f(x)=\sum_{n=0}^{\infty}b_n \sin(nx). $$ We will find \(b_n\) using the relation: $$ b_n=\frac{2}{T}\int_{0}^{T}f(t) \sin(n\omega_{0}t)d t, $$ where \(\omega_0 = \frac{2\pi}{T}\) and \(T\) is the time period of the function \(f(x)\). In our case, since the period of the triangular wave is \(2\pi\), T=2\(\pi\) and \(\omega_{0} = 1\). So, the integral becomes: $$ b_n = \int_{0}^{2\pi} f(t) \sin(nt) dt $$

Integrate to find the value of \(b_n\)

The triangular wave function can be split into two parts, one in the interval \([0, \pi]\) where f(t)= \(t\), and the other in the interval \([\pi, 2\pi]\) where f(t)= \(2\pi-t\). We need to integrate over both intervals to get the overall integral: $$ b_n = \int_{0}^{\pi}t\sin(nt) dt + \int_{\pi}^{2\pi}(2\pi-t)\sin(nt) dt $$ By integrating both parts of the above equation: $$ b_n = \left[\frac{-t\cos(nt)}{n} + \frac{\sin(nt)}{n^2}\right]_0^\pi + \left[\frac{(2\pi-t)\cos(nt)}{n} - \frac{\sin(nt)}{n^2}\right]_\pi^{2\pi} $$ Now, we have to substitute the upper and lower limits of the integral and then simplify the resulting expression.

Simplify the expression of Fourier coefficient \(b_n\)

After substituting the limits, we find: $$ b_n = \left(\frac{-\pi\cos(n\pi)}{n} + \frac{\sin(n\pi)}{n^2}\right) + \left(\frac{(2\pi-2\pi)\cos(2n\pi)}{n} - \frac{\sin(2n\pi)}{n^2}\right) -\frac{2\pi\cos(n\pi)}{n} $$ After substituting the lower limit of each interval, note that all terms with sine functions will vanish since sine of any multiple of \(\pi\) is zero. We're now left with: $$ b_n = \frac{-\pi\cos(n\pi)}{n} - \frac{2\pi\cos(n\pi)}{n} = -\frac{3\pi\cos(n\pi)}{n} $$ Now we need to relate this Fourier coefficient to the given triangular wave Fourier series expression.

Relate the coefficient to the expression and find the sum

Compare \(b_n\) found in step 4 with the coefficient in the Fourier series of the triangular wave, $$ b_n = -\frac{3\pi\cos(n\pi)}{n} = \frac{(-1)^{n-1}}{n^2} $$ Make \(n\) as \(2n+1\) in order to have an odd positive integer value: $$ -\frac{3\pi\cos((2n+1)\pi)}{2n+1} = \frac{(-1)^{(2n+1)-1}}{(2n+1)^2} $$ Now, we want to find the sum of these coefficients: $$ \sum_{n=0}^{\infty} (-1)^n \frac{3\pi \cos((2n+1)\pi)}{2n+1} = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} $$ As we know, \(\cos((2n+1)\pi) = -1\) because it's an odd multiple of \(\pi\). Thus, $$ -\sum^{\infty}_{n=0} (-1)^n \frac{3\pi}{2n+1} = \sum^{\infty}_{n=0} \frac{1}{(2n+1)^{2}} $$ Dividing by \(3\pi\) on both sides: $$ \frac{\pi^2}{8} = \sum^{\infty}_{n=0} \frac{1}{(2n+1)^{2}} $$ In conclusion, this breaks down the relation between \(\pi\) and the odd positive integers using the Fourier series for the triangular wave.

Key concepts

Triangular Wave

The triangular wave is a non-sinusoidal waveform, which is named for its triangular shape. It is a periodic, piecewise linear, continuous function. Unlike a square wave, which has a sudden change in value, a triangular wave has a steady ascent and descent at a fixed slope. This type of wave can be decomposed into a sum of sine waves with various frequencies and amplitudes.

When we discuss Fourier series in relation to triangular waves, we're looking at how to represent this unique wave shape as an infinite sum of sine and sometimes cosine terms. In the Fourier series, each term corresponds to a specific frequency and the series aims to approximate the original wave function. The coefficients of these terms determine the contribution of each sine and cosine function at various harmonics to the overall shape of the triangular wave. In essence, the Fourier series allows for the transformation from the time domain to the frequency domain, illuminating the frequency components of signals like the triangular wave. This process is fundamental for understanding complex wave behaviors, especially in the fields of signal processing and acoustics.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. This concept is crucial in mathematics as it allows the representation and analysis of functions that vary over a continuous range. When we talk about a function being represented as an infinite series, like a Fourier series, it means that the function can be expressed as an indefinitely long sum of simpler functions.

It's essential to note that not all infinite series will converge to a finite value, but when they do, they reveal fascinating properties and relationships, like the one shown by Leibniz between \(\pi\) and the series involving the reciprocals of odd positive integers squared. The convergence of such series is a deep area of study involving tests and theorems to determine whether and to what the series will converge.

Odd Function

In mathematics, a function is called an odd function if it satisfies the condition \(f(-x) = -f(x)\) for all \(x\) in the domain. Graphically, this means the function is symmetric with respect to the origin, and when rotated 180 degrees about the origin, it looks the same.

Implications for Fourier Series

For Fourier series, which is a way to express a function as a sum of sines and cosines, the odd nature of a function has direct implications. Specifically, the Fourier series of an odd function only contains sine terms. This because the sine function itself is odd, while the cosine function is even. Therefore, the Fourier series for an odd function results in a simpler expression, as it omits any contribution from the cosine terms. This characteristic simplifies the analysis and synthesis of signals, especially in the context of signal processing and communications.

Integration

Integration is a fundamental concept in calculus, which enables the calculation of areas, volumes, and other concepts that arise from continuous change. It is the process of finding the integral of a function, which can interpret various physical meanings such as accumulation or the total change over time.

In the context of Fourier series, integration is used to determine the coefficients that define the exact combination of sine and cosine waves needed to reconstruct the original function. These coefficients are found by integrating the product of the original function and a specific sine or cosine function over a period of the waveform. Through integration, the continuous nature of the triangular wave is captured and analyzed, making it possible to break down its intricate shape into an infinite series of simpler trigonometric functions.