Euler's constant, Taylor's formula, and slowly converging series

Taylor's formula is used to derive the Euler-Maclaurin summation formula for a more accessible study of infinite series. The former formula is a virtual corollary of the latter when the error in the midpoint rule for integration is estimated. This can be achieved by replacing a sum with an integral, resulting in a good estimate because of a small derivative and a bounded step size. Thus, this is a reserval of the standard effect. The Cauchy form of the remainder in Taylor's formula also determines the Euler-Maclaurin formula's error term. This is incidentally related to Bernoulli numbers.