An Alternative Approach to Computing $\beta (2k+1)$

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta (2k+1)$, 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 $\beta (2k)$. 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 $\zeta (2k)$.