What is another infinite series summation for Pi^2/6 besides 1/n^2?

In summary, the conversation discussed finding infinite series summations for Pi^2/6, specifically those that are not of the 1/n^2 family. The thread mentioned different series, such as 2(-1)^(n+1)/n^2 and 4/(2n)^2, and discussed their similarities to the 1/n^2 series. The conversation also touched on the difference between this thread and a previous one about finding infinite series for 1/5 and 1/7. Finally, the idea of multiplying a series for 1/5 by 5Pi^2/6 was mentioned as another approach. Overall, the conversation highlighted the enjoyment and excitement of finding new series for Pi^2/
  • #1
mesa
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So the title pretty much says it all, what other infinite series summations do we have for Pi^2/6 besides,

$$\sum_{n=1}^{\infty} 1/n^2$$

***EDIT*** I should also include,

$$\sum_{n=1}^{\infty} 2(-1)^(n+1)/n^2$$
$$\sum_{n=1}^{\infty} 4/(2n)^2$$
etc. etc.

A unique form outside of the 1/n^2 family.
 
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  • #2
http://www.wolframalpha.com/input/?i=Pi^2%2F6
 
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  • #4
dipole said:
http://www.wolframalpha.com/input/?i=Pi^2%2F6

Interesting, all of their series still have a 'squared' term for the denominator in some form or another. Do you know of any outside of the 1/n^2 family?

Either way thanks for the link!

Office_Shredder said:
How is this thread different than the one you posted before?

https://www.physicsforums.com/showthread.php?t=730791

That thread was about finding infinite series summations for 1/5 and 1/7 which eventually led to a 'general solution' for all '1/k' fractions for infinite series. This thread is specifically for Pi^2/6 and finding infinite series that are not of the 1/n^2 family.

Hope this helps.
 
  • #5
mesa said:
That thread was about finding infinite series summations for 1/5 and 1/7 which eventually led to a 'general solution' for all '1/k' fractions for infinite series. This thread is specifically for Pi^2/6 and finding infinite series that are not of the 1/n^2 family.

Hope this helps.

Take any series that equals 1/5 and multiply it by [itex] \frac{ 5 \pi^2}{6} [/itex].
 
  • #6
Office_Shredder said:
Take any series that equals 1/5 and multiply it by [itex] \frac{ 5 \pi^2}{6} [/itex].

Yes, that would do it too...
Do you have anything else besides Boreks standard answer on these things? :)

On a more serious note (sort of), deriving these new series is a blast! I have not encountered any other subject in mathematics that has been more fun!
 
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1. What is the Riemann zeta function?

The Riemann zeta function is an infinite series summation that is closely related to the sum of the reciprocals of the squares. It is defined as ζ(s) = 1/1^s + 1/2^s + 1/3^s + ... , where s is a complex variable. When s=2, the Riemann zeta function gives the sum of the reciprocals of the squares, which is equivalent to π^2/6.

2. Can you provide an example of another infinite series summation for π^2/6?

Yes, one example is the Basel problem solution, which states that the infinite series summation 1/1^2 + 1/2^2 + 1/3^2 + ... is equal to π^2/6. This is known as the Basel problem because it was first posed by the mathematician Pietro Mengoli in the 17th century in Basel, Switzerland.

3. How is the infinite series summation for π^2/6 related to the sum of the reciprocals of the cubes?

The infinite series summation for π^2/6 is not directly related to the sum of the reciprocals of the cubes. However, there is a formula called the Euler-Mascheroni constant, which is defined as the limit of the difference between the harmonic series and the natural logarithm function as n approaches infinity. This constant is closely related to the sum of the reciprocals of the cubes and can be used to derive a similar formula for π^3/20.

4. Are there any other infinite series summations for π^2/6?

Yes, there are other infinite series summations for π^2/6, such as the Gregory series and the series involving the square of the Catalan numbers. However, these series are not as well-known or commonly used as the Riemann zeta function or the Basel problem solution.

5. Why is the infinite series summation for π^2/6 important?

The infinite series summation for π^2/6 is important because it is linked to many other mathematical concepts, such as the Riemann zeta function, the Basel problem, and the Euler-Mascheroni constant. It also has practical applications in fields such as physics and engineering, where it is used to calculate the energy of certain systems and to model the behavior of materials.

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