Euler Series question and solution

Question:

Mengoli had posed the following series to be evaluated:

1+ \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \ldots.

Some great mathematicians, including Liebnitz, John Bernoulli and D’Alembert, failed to compute this infinite series. Euler established himself as the best mathematician of Europe (in fact, one of the greatest mathematicians in history) by evaluating this series initially by a not-so-rigorous method. Later on, he gave alternative and more rigorous ways of getting the same result.

Prove that the series converges and gets an upper limit. Then, try to evaluate the series.

Proof:

Due Nicolas Oresine:

Consider the following infinite series: \phi(s)=1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + ldots

We can re-write the preceding series as follows: \phi(s) = 1+ (\frac{1}{2^{s}}+\frac{1}{3^{s}}) + (\frac{1}{4^{s}} + \frac{1}{5^{s}} + \frac{1}{6^{s}} + \frac{1}{7^{s}}) + \ldots, which in turn is less than

1 + (\frac{2}{2^{s}}) + (\frac{4}{4^{s}}) + \ldots. Now, the RHS of this can be re-written as

1+(\frac{2}{2^{s}}) + (\frac{4}{4^{s}}) + \ldots=1 + \frac{1}{2^{(s-1)}}+ (\frac{1}{2^{(s-1)}})^{2} + \ldots, which is a geometric series and it is given by

\frac{1}{1-\frac{1}{2^{(s-1)}}}.

Now, we can say that \phi(s) will converge if \frac{1}{2^{(s-1)}}<1 \Longrightarrow s >1.

In order to prove what is asked, we start with \phi(s)=1 + \frac{1}{2^{s}}+ \frac{1}{3^{s}}+ \frac{1}{4^{s}}+\ldots

And, then multiply both sides by \frac{1}{2^{s}} and then subtract the resulting equation from the preceding equation to get

(1-\frac{1}{2^{2}})\phi(s)=1+\frac{1}{3^{s}}+\frac{1}{5^{s}}+\ldots

where all the terms containing the reciprocals of the sth power of even numbers vanished.

Repeating this procedure with \frac{1}{3^{s}} gives

(1-\frac{1}{2^{s}})(1-\frac{1}{3^{s}})\phi(s)=1+\frac{1}{5^{s}}+ \ldots

where all terms containing the reciprocals of the sth power of multiples of 3 vanished.

By continuing this with all prime numbers, we get

\prod_{p}(1-\frac{1}{p^{s}})\phi(s)=1, where p represents all prime numbers. Thus, we get

\phi(s)=1 + \frac{}{} + \frac{}{} + \frac{}{} + \ldots =\frac{1}{\prod_{p}(1-\frac{1}{p^{s}})}

This is a remarkable result because the LHS is concerned with only positive integers, whereas the RHS is concerned with only primes. This result is known as the “Golden Key of Euler”.

Riemann created his famous \zeta- function by extending the variable s to the entire complex plane, except s=1 withย 

\zeta(s)=1+ \frac{1}{2^{s}} + \frac{1}{3^{s}} + \ldots .

This function is now very famous as the Riemann zeta function.

How can we apply the Golden Key of Euler to Mengoli’s question that we started with?

Ans. In the Golden Key of Euler, substitute s=2.

Hence, we get the upper limit of the given series is 2.

Euler’s proof (1775):

The proof ran as follows:

It is a little roundabout way of arriving at the correct answer from a known result. Consider McLaurin’s series expansion of sin x:

\sin{(x)}=x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} -\frac{x^{7}}{7!} + \frac{x^{9}}{9!} + \ldots

By dividing both sides by x and then substituting y=x^{2} on the right side, we get the following:

\frac{\sin{(x)}}{x} = 1-\frac{y}{3!} + \frac{y^{2}}{5!} - \frac{y^{3}}{7!} + \ldots

By taking a special value of x=n\pi (and, hence y=n^{2}\pi^{2}), we get the following:

\frac{\sin (n\pi)}{(n\pi)}=0=1-\frac{y}{3!} + \frac{y^{2}}{5!} - \frac{y^{3}}{5!}+ \ldots

Noteย  that preceding equation is not a polynomial, but an infinite series.ย But, Euler still treated it as a polynomial (that is why it was not accepted as a rigorous result) and observed that this “infinite” polynomial has roots equal to y_{n}=n^{2}x^{2}. Then, Euler had used the fact that the sum of the reciprocals of the roots is determined by the coefficient of the linear term (here, the y-term) when the constant is made unity. (check this as homework quiz, for a quadratic to be convinced).ย So, Euler had arrived at the following result:

1-\sum_{i=1}^{\infty}\frac{6}{y_{n}}=0 \Longrightarrow \sum_{i=1}^{\infty}\frac{1}{y_{n}}=\frac{1}{6}. With y_{n}=n^{2}(\pi)^{2}, we get the following:

\sum_{i=1}^{\infty}\frac{1}{n^{2}(\pi)^{2}}=\frac{1}{6} or, \sum_{1}^{n^{2}}\frac{1}{n^{2}}=\frac{(\pi)^{2}}{6}.

Another proof also attributed to Euler that uses the series expansion of sin (x) goes as follows below:

\sin {(x)} has roots given by 0, \pm \pi, \pm 2\pi, \pm 3\pi, …So does this polynomial that Euler reportedly constructed:

x(1-\frac{x^{2}}{(\pi)^{2}})(1-\frac{x^{2}}{(2\pi)^{2}})(1-\frac{x^{2}}{(3\pi)^{2}})\ldots=0

So, Euler considered the preceding equation to be equivalent to:

\sin{(x)}=x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \ldots=0

Then, he had equated the coefficient of x^{3} in both to get the result:

\sum_{n=1}^{\infty}\frac{1}{n^{2}(\pi)^{2}} = \frac{1}{3!} = \frac{1}{6}.

Thus, \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{(\pi)^{2}}{6}.

Later on, Euler had provided a few more alternate and rigorous proofs of this result.

Reference: Popular Problems and Puzzles in Mathematics by Asok Kumar Mallik, IISc Press, Foundation Books.

Amazon India link:

https://www.amazon.in/Popular-Problems-Puzzles-Mathematics-Mallik/dp/938299386X/ref=sr_1_1?s=books&ie=UTF8&qid=1520930311&sr=1-1&keywords=popular+problems+and+puzzles+in+mathematics.

Hope you all enjoyed it — learning to think like Euler !! By the way, it did take a long time for even analysis to become so rigorous as it is now….You might like this observation a lot. ๐Ÿ™‚ ๐Ÿ™‚ ๐Ÿ™‚

Nalin Pithwa.

Is there a Fermat in you?? A solution possible

Reference: Popular Problems and Puzzles in Mathematics, Asok Kumar Mallik, Foundation Books, IISc Press.

Amazon India link:

https://www.amazon.in/Popular-Problems-Puzzles-Mathematics-Mallik/dp/938299386X/ref=sr_1_1?s=books&ie=UTF8&qid=1519414577&sr=1-1&keywords=Popular+problems+and+puzzles+in+mathematics

Fermat’s Problem:

Pierre de Fermat (1601-65), the great French mathematical genius, is described as the “Prince of Amateurs”, as he was not a professional mathematician and never published any work during his lifetime. As a civil servant, he served in the judiciary and spent his leisure with mathematics as his hobby. He corresponded with the best of mathematicians of his time. He teased the contemporary English mathematicians, Wallis and Digby, with the following problem, who had to admit defeat:

26 is a number that is sandwiched between a perfect square (25) and a perfect cube (27). Prove that there is no such number.

Solution:

We show that the equation y^{3}=x^{2}+2 has only one solution for positive integers x and y, viz., x=5 and y=3. First, we write y^{3}=x^{2}+2=(x+i\sqrt{2})(x-i\sqrt{2}). It can be shown that unique prime factorization occurs in the complex number system a+ ib\sqrt{2}. It can be also argued that since the product of the two complex numbers of that form is a cube, each factor must be a cube. So, we write:

x+i\sqrt{2}=(a+ib\sqrt{2})^{3}=a^{3}-6ab^{2}+i(3a^{2}b-2b^{3})\sqrt{2}.

Equating imaginary parts of both sides, we get

3a^{2}b-2b^{3}=b(3a^{2}-2b^{2})=1.

Thus, b=\pm 1 and 3a^{2}-2b^{2}=\pm 1, both having the same sign. But, with b=-1, one gets a^{2}=1/3, which is not possible. So, we take b=1 when a=\pm 1. Finally, with a=1, x=-5 and with a=-1, x=5. The sign of x is irrelevant as only x^{2} is involved in the original equation. With the solution x=5, we get y=3 as the only set of solution.

Cheers,

Nalin Pithwa.

Is there a “Fermat” in you? ? :-)

Fermat’s problem:

Pierre de Fermat (1601-65), the great French mathematical genius, is described as the “Prince of Amateurs” as he was not a professional mathematician and never published any work during his lifetime. As a civil servant, he served in the judiciary and spent his leisure with mathematics as his hobby. He corresponded with the best of mathematicians of his time. He teased the contemporary English mathematicians, Wallis and Digby, with the following problem, who had to admit defeat.

Problem:

26 is a number that is sandwiched between a perfect square (25) and a perfect cube (27). Prove that there is no other such number.

Please do try and let me know your solutions. Even partial solutions are welcome.

Cheers,

Nalin Pithwa.

Happy Numbers make Happy Programmers ! :-)

Here is one question which one of my students, Vedant Sahai asked me. It appeared in his computer subject exam of his recent ICSE X exam (Mumbai):

write a program to accept a number from the user, and check if the number is a happy number or not; and the program has to display a message accordingly:

A Happy Number is defined as follows: take a positive number and replace the number by the sum of the squares of its digits. Repeat the process until the number equals 1 (one). If the number ends with 1, then it is called a happy number.

For example: 31

Solution : 31 replaced by 3^{2}+1^{2}=10 and 10 replaced by 1^{2}+0^{2}=1.

So, are you really happy? ๐Ÿ™‚ ๐Ÿ™‚ ๐Ÿ™‚

Cheers,

Nalin Pithwa.