Looking for comprehensive list (or link) of even power summations

In summary, the conversation was about a function that could calculate exact values for some even powered summation series, but gave incorrect results for powers of 2 and 12+. The poster was looking for a comprehensive list of power summations in the form of $$\sum_{n=1}^{\infty} 1/n^m$$ where m is greater than the 10th power. Another user provided links to the Bernoulli numbers and their relationship to zeta values, which could be used to calculate the desired summations. The original poster then confirmed that the solution for n=12 provided by the links was correct, but their function gave a slightly different result. They planned to work on finding a solution from both
  • #1
mesa
Gold Member
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I have a function that results in 'exact' values for even powered summation series but it gives odd results for powers of '2' and '12+', how exciting! Unfortunately this also means the function is a far cry from a 'general solution'...

Does anyone have a comprehensive list of power summations in the form of,

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

where 'm' is greater than the 10th power in exact form? (even of course)
 
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  • #2
That's just [itex] \zeta(m)[/itex]. The Bernoulli numbers can be calculated explicitly and can be written in terms of zeta values, see

http://en.wikipedia.org/wiki/Bernoulli_numbers#Asymptotic_approximation

for the relationship between Bernoulli numbers and zeta values, and

http://en.wikipedia.org/wiki/Bernoulli_numbers#Explicit_definition

for a formula for calculating them. I also found this page

http://www.fullbooks.com/The-first-498-Bernoulli-Numbers.html

which says it has a list of the Bernoulli numbers but you might want to check that it is correct.

It's not clear what your post means when you say you have a function that results in exact values, but gives results that you think are wrong for almost all powers - in what manner is it supposed to be exact?
 
  • #3
Office_Shredder said:
It's not clear what your post means when you say you have a function that results in exact values, but gives results that you think are wrong for almost all powers - in what manner is it supposed to be exact?

The derived function will compute exact values of 'some' even powered summations. At the moment it will produce correct answers for,
$$\sum_{n=1}^{\infty} \frac{1}{n^4}=\frac{Pi^4}{90}$$
$$\sum_{n=1}^{\infty} \frac{1}{n^6}=\frac{Pi^6}{945}$$
$$\sum_{n=1}^{\infty} \frac{1}{n^8}=\frac{Pi^8}{9450}$$
$$\sum_{n=1}^{\infty} \frac{1}{n^{10}}=\frac{Pi^{10}}{93555}$$

But for powers of '12' and higher the results don't look right and I know '2' is also incorrect (It is giving Pi^2/20, not Pi^2/6). I can easily fix the '2' but I need to see what is going on with the other end since it is best to work both sides of the problem.

I could post the function although it is incomplete as far as being a 'general solution'. These discrepancies can likely be fixed but in order to do so I need a more comprehensive list of exact values for these power summations.

On another note, thanks for the links!
 
  • #4
Office_Shredder said:
That's just [itex] \zeta(m)[/itex]. The Bernoulli numbers can be calculated explicitly and can be written in terms of zeta values, see

http://en.wikipedia.org/wiki/Bernoulli_numbers#Asymptotic_approximation

for the relationship between Bernoulli numbers and zeta values, and

http://en.wikipedia.org/wiki/Bernoulli_numbers#Explicit_definition

for a formula for calculating them. I also found this page

http://www.fullbooks.com/The-first-498-Bernoulli-Numbers.html

which says it has a list of the Bernoulli numbers but you might want to check that it is correct.

Hey Office_Shredder, I had a chance to run through these links although I am not clear on what this has to do with the question I posed, am I missing something or was my post just a little too hazy and that led you off track?
 
  • #5
The first link gives the formula
[tex] B_{2m} = (-1)^{m+1} \frac{2 (2m)!}{(2\pi)^{2m}} \left( \sum_{n=1}^{\infty} 1/n^{2m} \right)[/tex]
which can be solved to give
[tex] \sum_{n=1}^{\infty} 1/n^{2m} = (-1)^{m+1} B_{2m} \frac{ (2\pi)^{2m}}{2(2m)!} [/tex]

So to calculate your sum on the left you need to evaluate the thing on the right, which is easy except for the B2m[/sum] part. I gave a link that will let you explicitly calculate it as a formula (if you wanted to compare your formula to the formula for the B's) and also a link which gives numerical evaluations of the first couple hundred.

For example, from the last link B36 = -26315271553053477373/1919190, so letting m=18 above,
[tex] \sum_{n=1}^{\infty} 1/n^{36} = 26315271553053477373/1919190 * \frac{ (2\pi)^{36}}{2(36)!} [/tex]

We can confirm these are in fact the same number.

http://www.wolframalpha.com/input/?i=zeta(36)
http://www.wolframalpha.com/input/?i=+26315271553053477373/1919190+*+\frac{+(2\pi)^{36}}{2(36)!}
 
  • #6
Office_Shredder said:
The first link gives the formula
[tex] B_{2m} = (-1)^{m+1} \frac{2 (2m)!}{(2\pi)^{2m}} \left( \sum_{n=1}^{\infty} 1/n^{2m} \right)[/tex]
which can be solved to give
[tex] \sum_{n=1}^{\infty} 1/n^{2m} = (-1)^{m+1} B_{2m} \frac{ (2\pi)^{2m}}{2(2m)!} [/tex]

So to calculate your sum on the left you need to evaluate the thing on the right, which is easy except for the B2m[/sum] part. I gave a link that will let you explicitly calculate it as a formula (if you wanted to compare your formula to the formula for the B's) and also a link which gives numerical evaluations of the first couple hundred.

For example, from the last link B36 = -26315271553053477373/1919190, so letting m=18 above,
[tex] \sum_{n=1}^{\infty} 1/n^{36} = 26315271553053477373/1919190 * \frac{ (2\pi)^{36}}{2(36)!} [/tex]

We can confirm these are in fact the same number.

http://www.wolframalpha.com/input/?i=zeta(36)
http://www.wolframalpha.com/input/?i=+26315271553053477373/1919190+*+\frac{+(2\pi)^{36}}{2(36)!}


Okay, it all makes sense now. I didn't notice in that first link to wikipedia that it was a summation for the far right term. Sometimes being so focused it is hard to spot even the obvious...

I am pleased to see that,
[tex] \sum_{n=1}^{\infty} 1/n^{12} = \frac{ 691Pi^{12}}{638512875} [/tex]

I 'threw out' my solution as being anomalous because it had a 'non 1' numerated fraction for an answer (just like this does). I unfortunately left my notebook at home (probably crumpled up buried under a pile of sheets and pillows :P) although I will check this once I get home.

So if this is in fact a 'general solution' then it is also a unique version of this Riemann Zeta function that doesn't require an input of your Bernoulli numbers, neat! (That is of course assuming the solutions for powers of 12+ are correct...)
 
  • #7
Office_Shredder, okay I checked it. The correct answer for n=12 is,

$$\sum_{n=1}^{\infty} 1/n^{12} = \frac{ 691Pi^{12}}{638512875}$$

My function gives,

$$\sum_{n=1}^{\infty} 1/n^{12} = \frac{ 733Pi^{12}}{638512875}$$

Close but not correct. I will work this from both ends and post when I have a solution. Thanks for the help.
 

1. What is an even power summation?

An even power summation is a mathematical concept in which a series of terms are added together, where each term is raised to an even power. These terms can be numbers, variables, or expressions. The resulting sum is called an even power summation.

2. What is the formula for an even power summation?

The formula for an even power summation is Sn = a2 + b2 + c2 + ... + n2, where n represents the number of terms in the summation and a, b, c, ... n represent the terms being added.

3. How is an even power summation different from an odd power summation?

An even power summation only includes terms raised to even powers (2, 4, 6, etc.), while an odd power summation includes terms raised to odd powers (1, 3, 5, etc.). Additionally, the formula for an odd power summation includes a coefficient in front of each term, while the formula for an even power summation does not.

4. What are some real-world applications of even power summations?

Even power summations are commonly used in physics, engineering, and statistics to model and analyze various systems and phenomena. For example, they can be used to calculate the kinetic energy of an object, determine the stability of a structure, or analyze the distribution of data in a population.

5. Is there a comprehensive list or link of even power summations available?

Yes, there are various resources available online that provide comprehensive lists of even power summations, including formulas, examples, and applications. Some good places to start include math websites such as Khan Academy, Mathworld, and MathHelp, as well as textbooks on algebra, calculus, and statistics.

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