TITLE:
The Connection between the Basel Problem and a Special Integral
AUTHORS:
Haifeng Xu, Jiuru Zhou
KEYWORDS:
Basel Problem, Zeta Function, Bernoulli Numbers, Bernoulli Polynomials
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.16,
September
4,
2014
ABSTRACT: By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, we find that this special integral is two times of another special integral. By using this fact we give an easy way to calculate the value of the alternating sum of without using the Fourier expansion. Also, we discuss the relationship between Genocchi numbers and Bernoulli numbers and get some results about Bernoulli polynomials.