sparknotes
The Taylor Series
Terms
Taylor's Formula
-
A formula expressing a function in terms of a polynomial approximation and an error
(remainder) term. Explicitly, given a function
f (x)
and a real number
a
,
where p n-1(x) is a Taylor polynomial and r n(x) is the remainder term
| f (x) | = |
f (a) + f'(a)(x - a) + ... + f
(n-1)(a) + 
f
(n)(t)
dt
|
|
| = | p n-1(x) + r n(x) |
where p n-1(x) is a Taylor polynomial and r n(x) is the remainder term
Taylor Polynomial
-
The approximation of a function
f (x)
around a point
x = a
by a polynomial
for some n≥ 0 .
p
n(x) = f (a) + f'(a)(x - a) + 
f''(a)(x - a)2 + ... + 
f
(
n)(a)(x - a)n
|
for some n≥ 0 .
Taylor Series
-
Given a function
f (x)
, the Taylor series about
x = a
is
The Taylor polynomials for f about a are the partial sums of this series.
 (x - a)n
|
The Taylor polynomials for f about a are the partial sums of this series.
Remainder Term
-
The difference between a Taylor polynomial and a function it approximates.
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