Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 76 / 2 / ii / Solution

For positive integer $n$ and any positive integer $N$, the finite geometric series identitygives an exact decomposition along the indented contour. The polynomial terms have no pole and may be returned to the positive real axis; their Gamma function integrals areEquivalently, expansion about the endpoint $t=0$ in the Laplace transform integral gives the same coefficients. The resulting formal asymptotic expansion isThe ratio of consecutive terms is $r/\lambda$, so the infinite series is factorially divergent. It should be truncated, not treated as a convergent sum; the pole prescription controls the contribution beyond its algebraic orders. Although the source initially calls $n$ an integer, the ordinary lower endpoint integral requires $n>0$, as is already implicit in the large-positive-$n$ limit of part (i).