I have tried, for long, to prove Taylor’s Theorem on my own. The only way that this proof is different from the hundreds of proofs online is that I have written as . This solves a lot of the problems I was facing in developing the Taylor expansion.
Let . Then we have
We can easily calculate that . Also,
Continuing in this fashion, we get that
Assuming that is smooth, if as , then we can recover the standard Taylor expansion formula.