Example 1

Let f(x) be a polynomial in x - c,

f(x) = a₀ + a₁(x - c) + ... + a_n(x - c)^n.

This is just a power series with all but the first n + 1 coefficients equal to zero. So by Theorem 1, the Taylor series of the polynomial is just the polynomial itself followed by infinitely many zeros,

a₀ + a₁(x - c) + ... + a_n(x - c)ⁿ + 0 + 0 + ....

We can also see this directly from Lemma 1 of the last section, namely

for m ≤ n.