A tale of two series – a Dickens of an integral

It is very often a straightforward matter to establish that an infinite series is convergent, but an entirely separate (and potentially complex) exercise to determine the sum of the series. Those series which I feel most comfortable with are alternating series of the form a1−a2+a3−a4+ · · · ,where the terms an ≥ 0. The reason for this is that if the terms decrease monotonically and tend to zero, i.e., an+1 ≤ an and an → 0 as n → ∞, then the series will converge. The remaining issue, therefore, is to determine the sum of the series. One of the most common examples in this class is the alternating harmonic series