Six Ways to Sum A Series

Imagine taking a square and dividing it along the diagonal. Now take one section and divide that down the diagonal. Repeat this process into infinity. This is a simple example of summing of an infinite series. Although this example provides and easily determined sum of the infinite series, the whole square, we find it commonly harder to find the sum of infinite series. In this paper, the author discusses the infinite series of the squares of the reciprocals of integers and different methods of finding its sum.
Found first by Leonhard Euler in 1734, the sum of this series is described today in a variety of ways, all of which are more mathematically acceptable than Euler’s original proof.

$\textbf{Euler’s Proof}$
The basic idea of Euler’s proof is to obtain a power series expansion for a function whose roots are multiples of the perfect squares. We can then apply a property of the polynomials to obtain the sum of the reciprocals of the roots. Here we represent the sine function as a power series:

(1)
\begin{align} \sin x = x - \frac{x^{3}}{3 \cdot 2} + \frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2} - \frac{x^{7}}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2} +... \end{align}

We can think of this expansion as an infinite polynomial. If we divide both sides by $x$, we obtain a polynomial with only the even powers of $x$. Once you replace $x$ with $\sqrt{x}$ the results is:

(2)
\begin{align} \frac{\sin \sqrt{x}}{\sqrt{x}} = 1 - \frac{x}{3 \cdot 2} + \frac{x^{2}}{5 \cdot 4 \cdot 3 \cdot 2} - \frac{x^{3}}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2} +... \end{align}

This is a function $f$ that has $\pi ^{2}, 4\pi ^{2}, 9\pi ^{2},$… for roots. Euler then used the fact that adding up the reciprocal of all the roots of a polynomial results in the negative of the ratio of the linear coefficient to the constant coefficient. Basically, if

(3)
\begin{equation} (x-r_{1})(x-r_{2})…(x-r_{n})=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} \end{equation}

then

(4)
\begin{align} \frac{1}{r_{1}}+\frac{1}{r_{2}}+…\frac{1}{r_{n}}=\frac{-a_{1}}{a_{0}} \end{align}

Euler assumed this same rule applied to power series and applied it to our function $f$.

(5)
\begin{align} \frac{1}{6}=\frac{1}{\pi^{2}}+\frac{1}{4\pi ^{2}}+\frac{1}{9\pi ^{2}}… \end{align}

If we multiply both sides of this equation by $\pi ^{2}$ we get $\frac{\pi ^{2}}{6}$ as the sum of the series. The problem with this is that power series are not polynomials and they do not share the same properties as polynomials. This means that we cannot always apply this rule to power series, but it doesn’t mean we can’t ever apply it. For this reason, however, Euler’s proof is said not to hold up to today’s proof standards.

$\textbf{Trignometry and Algebra}$

This proof method uses a special trigonometric identity which involves the angle $\omega = \frac{\pi}{(2m + 1)}$ and several of its multiples. The identity is

(6)
\begin{align} \cot ^{2} \omega + \cot ^{2} (2\omega ) + \cot ^{2} (3\omega ) +…+\cot ^{2} (m\omega ) = \frac{m(2m-1)}{3} \end{align}

We know that for any $x$ between $0$ and $\frac{\pi}{2}$, the following is true:

(7)
\begin{align} \sin x < x <\tan x \end{align}

Squaring and inverting this leads to

(8)
\begin{align} \cot ^{2} x < \frac{1}{x^{2}}<1 + \cot ^{2} x \end{align}

Now, using (6) we can successively replace x with $\omega, 2 \omega, 3 \omega,$ etc. This gives

(9)
\begin{align} \cot ^{2} \omega + \cot ^{2} (2 \omega ) + \cot ^{2} (3 \omega ) + … + \cot ^{2}(m \omega ) \end{align}
(10)
\begin{align} < \frac{1}{ \omega ^{2}} + \frac{1}{4 \omega ^{2}} + \frac{1}{9 \omega ^{2}} + … + \frac{1}{m^{2} \omega ^{2}} \end{align}
(11)
\begin{align} < m + \cot ^{2} \omega + \cot ^{2} (2 \omega ) + \cot ^{2} (3 \omega ) + … + \cot ^{2}(m \omega ) \end{align}

Using the identity ([[eerf label1]]) we have the equation

(12)
\begin{align} \frac{m(2m-1)}{3} < \frac{1}{\omega ^{2}}(1 + \frac{1}{4} + \frac{1}{9} + … + \frac{1}{m ^{2}} < \frac{m(2m-1)}{3} + m \end{align}

Finally we can substitute $\omega = \frac{\pi}{(2m+1)}$ for:

(13)
\begin{align} \frac{m(2m-1)\pi ^{2}}{3(2m+1)^{2}} < 1 + \frac{1}{4} + \frac{1}{9} + … + \frac{1}{m ^{2}} < \frac{m(2m-1)\pi ^{2}}{3(2m+1)^{2}} + \frac{m\pi ^{2}}{(2m+1)^{2}} \end{align}

This set of inequalities provides upper and lower bounds for the sum of the first m terms of Euler’s series. If we let $m$ go to infinity, the lower bound is

(14)
\begin{align} \frac{m(2m-1)\pi ^{2}}{3(2m+1)^{2}}= \frac{\pi ^{2}}{6}\frac{2m ^{2} – m}{2m ^{2} + 2m +0.5} \end{align}

This approaches $\frac{\pi ^{2}}{6}$. The upper bound also approaches $\frac{\pi ^{2}}{6}$, confirming Euler’s finding.

The remaining four proofs will not be presented in the same manner as above but rather will be summarized.

$\textbf{Odd Terms, Geometric Series, and a Double Integral}$

This proof involves separating the summation of the odd and even terms of the sequence. Defining $E=\Sigma ^{\infty} _{k=1} \frac{1}{k ^{2}}$ and then calculating the integral representation of the even terms of the sequence shows us that these terms make up one-fourth of the value of $E$. If we define the sum of the odd terms as three-fourths of this $E$, we can develop another integral to describe the odd terms as eventually we determine, once again, that the sum of the sequence is $\frac{\pi ^{2}}{6}$.

$\textbf{Residue Calculus}$

Using a concept in residue calculus of complex integrals it is possible to calculate the sum of the series in question. The function used is $f(z) = \frac{\cot(\pi z)}{z^{2}}$ and the path ($P_{n}$) is the rectangle centered at the origin with sides parallel to the real and imaginary axis in the complex plane. The sides of this rectangle intersect the real axis at $^{+}_{-} (n + \frac{1}{2})$ and the imaginary axis at $^{+}_{-}ni$. Carrying out the residue calculations using this function and path we conclude that the same sum is reached for our series.

$\textbf{Fourier Analysis}$

This proof uses some concepts in Fourier analysis and compares them to the series. In Fourier analysis, the dot product of two functions is defined as $f \cdot g = \frac{1}{2\pi}\int ^{\pi}_{-\pi} f(t)g(t)dt$. If we are describing the dot product of the a function an itself, we can also write $f \cdot f = ...|a_{-2}|^{2} + |a_{-1}|^{2} + |a_{0}|^{2} + |a_{1}|^{2} + |a_{2}|^{2} + ...$. If we define, $f(t)=t$ in terms of our $a_{n}$ coefficients, we discover that it is just our series written twice (if the sum is $E$ then this sum is $2E$) and when we compute the sum using our dot product integral formula it is $\frac{\pi ^{2}}{3}$. So $2E=\frac{\pi ^{2}}{3}$, therefore $E=\frac{\pi ^{2}}{6}$.

$\textbf{A Real Integral with an Imaginary Value}$

This proof begins with the integral $I=\int ^{\frac{\pi}{2}}_{0} \ln (2\cos x)dx$. The logarithm present (because $2 \cos (x) = e^{ix} + e^{-ix}$) is replaced with a power series and integration is performed term by term. This gives us the odd terms of our series which we know are three-fourths of the sum of the total. These terms are multiplied by $\frac{1}{i}$ which gives us $\frac{-3i}{4}E$. Substituting this into our original integral we get $I=i(\frac{\pi ^{2}}{8} - \frac{3}{4}E)$. Setting the right-hand side of this equation to $0$ we get our familiar answer.

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$\textbf{Connection to Real Analysis}$

This article is all about computing the sum of a sequence of real numbers. The fact that we are dealing with a sequence of real numbers places the topic of the article securely into the realm of Real Analysis. However, it also deals with different methods of computing this sum which is the real heart of analysis. It is easy to see from this article that some methods are easier than others are and all achieve the same end which is important in deciding which method works for the situation of proof you may be extending this result in to.

$\textbf{Context of the Article}$

The broader field of study for topics like the one covered in this article is sequences and series. Truly a topological study, series and sequences, and more specifically infinite sequences, are of particular interest to many because of the seemingly implausible ability to sum up the terms of an infinite series. For further reading on this subject, the best idea would to be to develop as sequence or series that interests the reader and research that particular series or read a real analysis textbook or other online article about summing infinite series.

Bibliography
Six Ways to Sum a Series
Dan Kalman
The College Mathematics Journal, Vol. 24, No. 5. (Nov., 1993), pp. 402-421.