This paper presents a new approach to evaluating the special values of the
Dirichlet beta function, β(2k+1), where k is any nonnegative integer.
Our approach relies on some properties of the Euler numbers and polynomials,
and uses basic calculus and telescoping series. By a similar procedure, we also
yield an integral representation of β(2k). The idea of our proof adapts
from a previous study by Ciaurri et al., where the authors introduced a new
proof of Euler's formula for ζ(2k).Comment: 11 page