SECTION 5.4 Polynomial Approximations 291
=
∑n
k=0f(k)(a)
n!(x−a)k.We expect, then, to have a pretty good approximation
f(x) ≈∑n
k=0f(k)(a)
n!
(x−a)k.The polynomial Pn(x) =∑n
k=0f(k)(a)
n!(x−a)k is called the Taylorpolynomial of degreenforf at x=a. Ifa= 0, the above polyno-
mial becomes
f(x) ≈∑n
k=0f(k)(a)
n!xkand is usually called theMaclaurin polynomial of degree nforf.
What if, instead of stopping at a degreenpolynomial, we continued
the process indefinitely to obtain a power series? This is possible and
we obtain
∑∞
k=0f(k)(a)
n!
(x−a)k Taylor seriesforf atx=a, and∑∞
k=0f(k)(a)
n!xk Maclaurin seriesforf.Warning. It is very tempting to assume that the Taylor series for a
functionf will actually converge tof(x) on its interval of convergence,
that is,
f(x) =∑∞
k=0f(k)(a)
n!(x−a)k.