We demonstrate a tight analysis of an expectation of a sum of exponents raised to some power, prevalent in, but not confined to, Gallager's bounding techniques. We show that the traditional analysis that uses Jensen's inequality, although tight in Gallager's random coding error exponent, might not be tight in general. Using the binary symmetric channel as an example, we show that R c — the lowest rate at which Gallager's bound agrees with the sphere packing bound is the lowest rate for which Jensen's inequality is tight for a range of possible parameters.