Unit 10 Notes: Taylor & Maclaurin Series (AP Calculus BC)

Taylor polynomial

A polynomial approximation of a function near a point x=a that matches the function’s value and derivatives at a up through a chosen order.

Center (a) of a Taylor polynomial/series

The x-value a about which the approximation is built; powers appear as (x−a)^k, and the polynomial is tuned to the function at x=a.

Degree-n Taylor polynomial (P_n or T_n)

The Taylor polynomial that includes terms through (x−a)^n, matching $f(a), f'(a), …, f^{(n)}(a)$.

Taylor polynomial formula

P_n(x)=$\frac{f^{(k)}(a)}{k!}$^k.

Maclaurin polynomial

A Taylor polynomial centered at a=0 (so terms are in powers of x^k).

Factorial (k!) in Taylor coefficients

The normalization factor in Taylor terms; the k-th term uses f^(k)(a)/k!, and forgetting k! is a common error.

Taylor series

The infinite extension of Taylor polynomials: ∑_{n=0}^{∞} $\frac{f^{(n)}(a)}{n!}$ (when it converges and represents the function).

Maclaurin series

A Taylor series centered at 0: $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}$ x^n.

Remainder (error) term R_n(x)

The difference between the function and its degree-n Taylor polynomial: R_n(x)=$f(x)−P_n(x)$.

Lagrange Error Bound

A guaranteed upper bound on Taylor approximation error: |R_n(x)| ≤ $\frac{M}{(n + 1)!} |x−a|^{(n + 1)}$, where M bounds |f^{(n + 1)}(t)| on the interval from a to x.

M in the Lagrange Error Bound

A number satisfying |f^(n+1)(t)| ≤ M for all t between a and the target x; it must be chosen as a valid maximum bound on that interval.

“Next derivative” in error bounds (f^(n+1))

For a degree-n Taylor polynomial, the Lagrange bound uses the (n+1)th derivative, not the nth derivative.

“Safe zone” for a Taylor approximation

The idea that Taylor polynomials approximate best near the center a; using them far from a without error reasoning can be unreliable.

Power series

An infinite series of the form ∑_{n=0}^{∞} c_n (x−a)^n, which may converge for some x-values and diverge for others.

Radius of convergence (R)

A number R such that a power series converges for |x−a|R; endpoints |x−a|=R require separate checks.

Interval of convergence

The set of x-values where a power series converges, typically (a−R, a+R) plus any endpoints that also converge after testing.

Endpoint check

The required step of testing convergence separately at x=a±R because behavior at endpoints can differ from interior points.

Ratio Test

A convergence test using L=\lim_{n o \infty} $\frac{|u_{n+1}|}{|u_n|}$: converges if L

Harmonic series

The series ∑_{n=1}^{∞} $\frac{1}{n}$, which diverges (often used in endpoint comparisons).

Alternating harmonic series

The series ∑_{n=1}^{∞} (-1)^n/n (or equivalent form), which converges (a common endpoint result).

Geometric series (standard form)

1/(1−x)=∑_{n=0}^{∞} x^n, valid for |x|

Term-by-term differentiation of a power series

If f(x)=∑{n=0}^{∞} cn x^n, then f'(x)=∑{n=1}^{∞} n cn x^(n−1) (valid on the interval of convergence; same radius of convergence).

Term-by-term integration of a power series

If f(x)=∑{n=0}^{∞} cn x^n, then ∫f(x)dx = C + ∑{n=0}^{∞} [cn/(n+1)] x^(n+1) (valid on the interval of convergence; same radius of convergence).

Standard Maclaurin series for ln(1+x)

ln(1+x)=$\frac{1}{(-1)^{n+1}}$ imes \frac{x^n}{n}, with convergence on $-1 < x \le 1$.

Standard Maclaurin series for arctan(x)

arctan(x)=$\frac{(-1)^n x^{2n + 1}}{(2n + 1)}$, valid for |x| \le 1 (endpoints included).