Making use of a newly developed package in the computer mathematics system SageMath, we show how to perform a full asymptotic analysis of certain types of sums that occur frequently in combinatorics, including explicit error bounds. We present two applications of the general approach to illustrate its use: the first concerns a classical problem due to Ramanujan, while the second one concerns a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.