I often tell my students that **Integration by Parts** is **Applied Math Trick #1**. I once had someone retort with “last year, you said Taylor Series was trick #1”.

Well, Taylor series is just really integration by parts… Forever! I was surprised that students hadn’t seen this before. It’s remarkably simple, and yields the remainder theorem quite nicely.

## Taylor Series via Integration by Parts

Beginning with the identity that

, a quick integration by parts where where we will let will give

.

Repeating this process again, we find…

.

If you keep going, you can see precisely how this will all play out. Furthermore, the error in truncating at some point is bounded by the integral! A simple estimate of the integral gives you exactly what you expect in terms of bounds on the integral.