Exploring the Basel Problem: Proving pi*pi/12

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In summary, the Basel problem states that the sum of the first n terms of a Fourier series converges to a constant value pi*pi/6.
  • #1
psholtz
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The Basel Problem is a well known result in analysis which basically states:

[tex]
\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}
[/tex]

There are various well-known ways to prove this.

I was wondering if there is a similar, simple way to calculate the value of the sum:

[tex]
\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ... = ?
[/tex]

The value of this sum should work out to pi*pi/12, but I was wondering if there was a straightforward way to prove it?
 
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  • #2
Depends on what you mean with straightforward. Do you think Fourier series is straightforward? That allows you to prove it.
 
  • #3
psholtz said:
The Basel Problem is a well known result in analysis which basically states:

[tex]
\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}
[/tex]

There are various well-known ways to prove this.

I was wondering if there is a similar, simple way to calculate the value of the sum:

[tex]
\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ... = ?
[/tex]

The value of this sum should work out to pi*pi/12, but I was wondering if there was a straightforward way to prove it?

Well, it may not be rigorous, but you could write

$$\begin{eqnarray*}
\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ...& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - 2\left(\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \dots \right) \\
& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - \frac{2}{2^2}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots \right) \\
& = & \dots\end{eqnarray*}$$

(You could make this rigourous by being careful with the convergence of the two series, I imagine).
 
  • #4
Sure, a Fourier series would be straightforward.

I'm familiar w/ how Fourier analysis can be used to sum the first series, but it's not immediately clear to me how to proceed from that solution, to the sum for the second series.

Could you give me a pointer/hint?
 
  • #5
Mute said:
Well, it may not be rigorous, but you could write

$$\begin{eqnarray*}
\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ...& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - 2\left(\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \dots \right) \\
& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - \frac{2}{2^2}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots \right) \\
& = & \dots\end{eqnarray*}$$

(You could this rigourous by being careful with the convergence of the two series, I imagine).
Yes, thanks...

This was something along the lines of the intuition I was going by, but didn't quite get it to this point.

Thanks..
 
  • #6
Mute said:
Well, it may not be rigorous, but you could write

$$\begin{eqnarray*}
\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ...& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - 2\left(\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \dots \right) \\
& = & \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\right) - \frac{2}{2^2}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots \right) \\
& = & \dots\end{eqnarray*}$$

(You could this rigourous by being careful with the convergence of the two series, I imagine).

Oh, this works as well! Nice! :smile:

My idea was to work with the Fourier series of [itex]f(x)=x^2[/itex]
 
  • #7
Darn, guys, you replied too fast and quoted my post before I could edit in the missing word "make". =P
 

FAQ: Exploring the Basel Problem: Proving pi*pi/12

1. What is the Basel Problem?

The Basel Problem, also known as the Basel Summation Problem, is a famous mathematical problem posed by Pietro Mengoli in 1644. It asks for the exact sum of the reciprocal of the squares of all the positive integers, also known as the Basel Sum. This problem has puzzled mathematicians for centuries and has led to the discovery of important mathematical concepts, such as the Riemann zeta function.

2. Why is proving pi*pi/12 important?

Proving that the Basel Sum is equal to pi squared divided by 12 has significant implications in number theory and other branches of mathematics. It also helps to understand the nature of the Riemann zeta function and its relation to other mathematical functions. Additionally, solving the Basel Problem adds to the body of knowledge in mathematics and contributes to the development of new theories and ideas.

3. How has the Basel Problem been approached in the past?

Many famous mathematicians, such as Leonhard Euler and Johann Bernoulli, attempted to solve the Basel Problem without success. Some used geometric and trigonometric methods, while others tried to find a pattern in the partial sums of the series. However, none of these approaches led to a conclusive proof.

4. What are some modern approaches to solving the Basel Problem?

In modern times, the Basel Problem has been tackled using advanced mathematical techniques, such as complex analysis and the theory of modular forms. Some researchers have also used computer algorithms and computational methods to find numerical solutions. However, a rigorous mathematical proof has yet to be found.

5. Why is proving the Basel Problem so challenging?

The Basel Problem is considered a challenging mathematical problem because it requires a deep understanding of various mathematical concepts, including complex analysis, number theory, and algebraic geometry. Moreover, the Basel Sum does not converge quickly, making it difficult to find patterns and make conjectures. Additionally, there is currently no known closed-form solution for the Basel Sum, which adds to the complexity of solving the problem.

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