I would like to know the behavior of the Riemann zeta function values at even and odd integers for studying irrationality between those values. I have tried using wolfram alpha to check the value of this sum: $$ \sum_{n = 1}^{\infty}\left[\zeta\left(2n\right)-\zeta\left(2n + 1\right)\right]. $$ It tells me it equals $\frac{1}{2}$ .
Note: I don't have any method to show if the titled sum is true . Maybe I find who is help me here for evaluating the titled sum.
Thanks for any help.