Taylor's Theorem: Explaining the Sum Telescoping

[glebovg]

http://home.iitk.ac.in/~psraj/mth101/lecture_notes/lecturer10.pdf

Can anyone explain (1)?

Is it because the sum telescopes?

 

[HallsofIvy]

Yes. By the product rule, the derivative of each term contains two parts: the part where you are differentiating $f^{(n)}(x)$ and the part where you re differentiating $(b- x)^n$. The first part cancels the previous term and the second part cancels the next term.