Take an elementary convergent integral like:
$\int^\infty_0 e^{- \lambda x} = \frac{1}{\lambda} $
If you series expand it every term and integrate term-by-term every term integrates to infinity. Is there a systematic way to cut-off the integral if you keep the $n^{th}$ term in the series so that you can reasonably approximate the integral to some quantified error?
EDIT: Clearly the series expansion is of little use in the above integral, but I am interested in a potential case where, for example, I find an integral that converges when I numerically integrate it but the analytical series diverges.