Let γn denote the value of
where n is a definite integer; and let γ denote the limit of
when the integer k is indefinitely increased. It is known that the expansion of γn – γ in ascending powers of 1/n is
where B1, B3, B5… are the numbers of Bernoulli. The series (3) is, however, divergent, as B2r+1 not only increases indefinitely with r, but bears† an infinite ratio to B2r–1 in this case. It is proposed to find by elementary methods the expansion of γn – γ up to the term in nr and to estimate the error (of order l/nr+1) made in omitting further terms of series (3). I shall take the case of r = 9, but the process is quite general.