Terry Tao shows how to get from side of equality to the other: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta

Terry Tao shows how to get from side of equality to the other: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri... It goes into detail and shows the derivation and internal consistency of the method.