Unit VI: Taylor Series
6.1 Introduction to Power Series
A power series is an infinite series of the form:
∑n=0∞an(x - c)n = a0 + a1(x-c) + a2(x-c)2 + ...
where c is the center and an are the coefficients. The radius of convergence R determines where the series converges.
6.2 Taylor Series
The Taylor series of a function f(x) about the point x = a is:
f(x) = ∑n=0∞ f(n)(a)/n! (x - a)n
= f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + f'''(a)(x-a)3/3! + ...
This expresses a smooth function as an infinite polynomial centered at a.
6.3 Maclaurin Series
A Maclaurin series is a Taylor series centered at a = 0:
f(x) = ∑n=0∞ f(n)(0)/n! xn
Important Maclaurin Series:
| Function | Series | Convergence |
|---|---|---|
| ex | 1 + x + x2/2! + x3/3! + ... | All x |
| sin x | x - x3/3! + x5/5! - ... | All x |
| cos x | 1 - x2/2! + x4/4! - ... | All x |
| 1/(1-x) | 1 + x + x2 + x3 + ... | |x| < 1 |
| ln(1+x) | x - x2/2 + x3/3 - ... | -1 < x ≤ 1 |
| (1+x)n | 1 + nx + n(n-1)x2/2! + ... | |x| < 1 |
6.4 Convergence of Taylor Series
The Taylor series converges if the remainder term Rn(x) approaches zero as n approaches infinity.
Taylor's Theorem with Remainder:
f(x) = Tn(x) + Rn(x)
where Tn(x) is the nth-degree Taylor polynomial and Rn(x) is the Lagrange remainder:
Rn(x) = f(n+1)(c)/(n+1)! (x-a)n+1
for some c between a and x.
6.5 Radius and Interval of Convergence
Use the ratio test to find the radius of convergence R:
R = limn→∞ |an/an+1|
The series converges for |x - a| < R and diverges for |x - a| > R. Check endpoints separately.
Example: Find the radius of convergence of ∑xn/n!
R = lim |an/an+1| = lim |(n+1)!/n!| = lim (n+1) = ∞
The series converges for all x (which makes sense since it equals ex).
6.6 Geometric Series
The geometric series is one of the simplest and most useful power series:
∑n=0∞rn = 1/(1-r), for |r| < 1
Partial sum: Sn = (1 - rn)/(1 - r)
Many functions can be expressed using geometric series by substitution:
- 1/(1+x) = 1 - x + x2 - x3 + ... (substitute r = -x)
- 1/(1-x2) = 1 + x2 + x4 + ... (substitute r = x2)
6.7 Applications of Taylor Series
Approximation: Taylor polynomials provide polynomial approximations to functions. The more terms used, the better the approximation near the center.
Computing Limits: Taylor series can evaluate limits that are difficult with other methods.
Example: limx→0 (sin x - x)/x3
sin x = x - x3/6 + ...
(sin x - x)/x3 = (-x3/6 + ...)/x3 = -1/6
Integration: Some integrals without closed forms can be computed via Taylor series term-by-term integration.
Example: ∫e-x²dx has no elementary antiderivative, but:
e-x² = 1 - x2 + x4/2! - x6/3! + ...
∫e-x²dx = x - x3/3 + x5/10 - x7/42 + ... + C
6.8 Taylor Series for Multivariable Functions
For f(x,y) about point (a,b), the Taylor expansion is:
f(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) + (1/2!)[fxx(a,b)(x-a)2 + 2fxy(a,b)(x-a)(y-b) + fyy(a,b)(y-b)2] + ...
The first-order approximation gives the tangent plane, while higher-order terms improve accuracy.
Example: Expand f(x,y) = excos y about (0,0) up to second order
f(0,0) = 1, fx = 1, fy = 0
fxx = 1, fxy = 0, fyy = -1
f(x,y) ≈ 1 + x + (1/2)(x2 - y2)