One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined by
Ramanujan (9) showed that when n is large, θn possesses the asymptotic expansion
The first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined by
then πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansion
A generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by
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