Taylor Series Derivation

Calculus 2 · Axiom Academy

LESSON Taylor Series Derivation Where does the formula come from? Assume a function is a power series, then let differentiation hand you every coefficient. 1. Assume the Function IS a Power Series Start by supposing f can be written as an infinite polynomial in powers of (x-a) , with unknown coefficients that we intend to solve for: The unknowns are the coefficients c_k Why is this reasonable? Truncating after the degree- n term gives the Taylor polynomial P_n(x) — and as you keep more terms, P_n hugs the true curve more tightly near x=a . The animation builds P_1 , then P_2 , then P_3 for f(x)=e^x at a=0 , each closing in on f . 2. Differentiate and Evaluate — Only One Term Survives Here is the whole trick. To isolate a single coefficient c_k , differentiate the assumed series k times , then plug in x = a . Watch what each term of does. The grid below is the engine of the proof: row j is the term c_j(x-a)^j , and each column applies one more derivative. Plugging in x=a kills every cell off the diagonal — a factor of (x-a) still remains (so it vanishes at a ), or the term was already differentiated to 0 . Only the diagonal cell j=k survives , and it equals . Differentiate k times, then set x=a : The factorial k! is exactly the constant the k -th derivative pulls out of (x-a)^k — it was never put in by hand; it falls out of differentiating a power k times. Sanity check on the lowest terms

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