Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 30A (Asymptotic Methods)

An asymptotic power series about $a$ means that for every fixed $N\ge 0$,as $x\to a$ through the specified domain. It makes no assertion that the infinite series converges at nonzero $x-a$. If a power series converges in a neighbourhood, choose a smaller fixed convergence radius $r$: absolute convergence bounds its tail after $N$ by $|x-a{|}^{N+1}{\sum }_{k>N}|c_k|r^{k-N-1}$ for $|x-a|\le r$. Thus a convergent power series is also asymptotic. The coefficients are unique, successively recovered byIf two expansions first differed at index $k$, division of their difference by $(x-a{)}^k$ would contradict this limit.

For the given integral, use the exact finite identity $1/(1+2xt)={\sum }_{k=0}^{n-1}(-2xt{)}^k+(-2xt{)}^n/(1+2xt)$. The gamma integral givesConsequentlyFor $t\ge 0$ the denominator is at least one, soThe empty product for $k=0$ is one. The bound is exactly the magnitude of the next omitted term. For the asymptotic definition through degree $N$, take $n=N+1$, giving an $O(t^{N+1})$ remainder. Only a finite expansion was integrated; interchanging a divergent infinite series with the integral would be invalid. Indeed the ratio of consecutive term magnitudes is $(2k+1)t$, so this formal series diverges at every fixed $t>0$.