nth-degree Taylor polynomial of f(x) about x = x0
Pn(x) = f(x0) + f′(x0)(x − x0) + f′′(x0)/2! (x − x0)^2 + · · · +f(n)(x0)/n! (x − x0)^n
nth-degree Taylor remainder
f(x) = Pn(x) + Rn(x)
Rn(x) = f(x) − Pn(x)
error associated with the approximation
|Rn(x)|
What are the conditions for Taylor’s Theorem?
f(x) must be continuous
f(x) must have n+1 derivatives on interval I containing x0
What is the general statement of Taylor’s Theorem?
For all x in I, there exists c between x0 and x such that: f(x) = Pn(x) + Rn(x)
What is the Taylor polynomial Pn(x)?
Pn(x) = f(x0) + f’(x0)(x - x0) + f’‘(x0)/2! (x - x0)^2 + … + f^(n)(x0)/n! (x - x0)^n
What is the remainder term Rn(x)?
Rn(x) = f^(n+1)(c)/(n+1)! (x - x0)^(n+1) where c lies between x0 and x
What does the remainder term Rn(x) represent?
The error between the actual function f(x) and the Taylor polynomial Pn(x)
Shows how well the polynomial approximates f(x)
What is the Maclaurin series?
The Taylor series when x0 = 0: f(x) = f(0) + f’(0)x + f’‘(0)/2! x^2 + … + f^(n)(0)/n! x^n + …
What is the difference between Taylor and Maclaurin series?
Taylor series: expansion about any point x0
Maclaurin series: special case when x0 = 0
What does the Taylor/Maclaurin series represent?
An infinite polynomial expansion that approximates f(x) near x0 (or near 0 for Maclaurin).
What is the general form of a power series about x = x0?
Σ (n=0 to ∞) an (x - x0)^n = a0 + a1(x - x0) + a2(x - x0)^2 + … + an(x - x0)^n + …
What is the Ratio Test
For series Σ bn, compute L = lim (n→∞) (bn+1 / bn).
If L < 1 → converges
If L > 1 → diverges
If L = 1 → inconclusive
How do we find the radius of convergence of a Taylor series
Radius R = lim (n→∞) (an / an+1). Series converges for |x - x0| < R. (where an = f^(n)(x0)/n!)
When does a Taylor series equal the original function
f(x) = Σ (n=0 to ∞) f^(n)(x0)/n! (x - x0)^n if and only if lim (n→∞) Rn(x) = 0. That is, the remainder tends to zero as n → ∞.
What is the binomial expansion
(1 + x)^α = 1 + αx + α(α - 1)/2! x^2 + … + α(α - 1)…(α - n + 1)/n! x^n + … Valid for |x| < 1
What is the Taylor series for a function of two variables
f(x, y) = f(x0, y0) + (x - x0)(∂f/∂x) + (y - y0)(∂f/∂y) + 1/2! [ (x - x0)^2 (∂²f/∂x²) + 2(x - x0)(y - y0)(∂²f/∂x∂y) + (y - y0)^2 (∂²f/∂y²) ] + … (All derivatives evaluated at (x0, y0))
