I have just had an interesting mathematical thought, which I do not know the answer to.
Here it is:
I was watching a documentary called " The Story of Maths" hosted by Marcus du Sautoy, a documentary about the history of mathematics, and Marcus said that Euler proved that the infinite series 1/1+1/4+1/9... equals (pi^2)/6.
( My mother thought this very odd.) Anyway, my thought was this:
Supposed that you calculated all the results of all of these kinds of series ( thus, you would be calculating 1/1+1/8+1/27... and 1/1+1/16+1/81... and so forth.) My question is, is there a pattern between all these results? And if so, can you use this pattern to calculate the result for the next infinite series? ( that is, if you have the pattern and also have the result for the series before it, can you calculate the result of the series after it?)
Hope you enjoy this problem!
You are talking about the Riemann zeta function with an integer parameter s. See:
ReplyDeletehttp://en.wikipedia.org/wiki/Riemann_zeta_function
If you look at:
http://en.wikipedia.org/wiki/Riemann_zeta_function#Specific_values
You'll see a definition for when s is even. It uses Bernoulli numbers:
http://en.wikipedia.org/wiki/Bernoulli_number
which can be calculated recursively:
http://en.wikipedia.org/wiki/Bernoulli_number#Recursive_definition
That is to say that if you know all of the first k-1 Bernoulli numbers, you can use them to calculate the kth Bernoulli number. The values you get for even ks can then be used to calculate the corresponding value of the Riemann zeta function.
Apparently no one has yet found such a formula for odd integers. Perhaps you will!