Sequences and Series, and new Coordinate Systems

Maclaurin Series

Taylor Series centered at 0

harmonic series

a series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency

Equation for Concavity

d^2y/dx^2 = ( (d/dt)(dy/dx) )/x'(t)

Taylor Series

f(c)+f'(c)(x-c)+f''(c)(x-c)^2/2 ... +f^n(c)(x-c)^n/n!

Polar Curves Symmetry ---> Symmetric with respect to x-axis if...

r(θ) = r(-θ)

Polar Curves Symmetry ---> Symmetric with respect to the origin if...

r(θ) = r(pi+θ)

Polar Curves Symmetry ---> Symmetric with respect to y-axis if...

r(θ) = r(pi-θ)

Rectangular ---> Polar Coordinates

r^2 = x^2 + y^2 tanθ = y/x

Geometric Series

the sum of the terms of a geometric sequence

Polar ---> Rectangular Coordinates

x = rcosθ y = rsinθ

f(x) = (1+x)^r, Expressed as a Taylor Series

Σ (r choose n)(x^n) = (r(r-1)-(r-(n-1)))/n!

f(x)=Ln(1+x), Expressed as a Taylor Series, and Convergence

Σ(-1)^(n+1) (x^n/n), converges (-1,1]

f(x) = cos(x), Expressed as a Taylor Series, and Convergence

Σ(-1)^n (x^(2n)/(2n)!), converges (-∞,∞)

f(x) = sin(x), Expressed as a Taylor Series, and Convergence

Σ(-1)^n (x^(2n+1)/(2n+1)!), converges (-∞, ∞)

f(x) = arctan(x), Expressed as a Taylor Series, and Convergence

Σ(-1)^n (x^(2n+1)/2n+1), converges [-1,1]

f(x) = e^x, Expressed as a Taylor Series, and Convergence

Σ(x^n)/n!, converges (-∞, ∞)

f(x) = 1/(1-x), Expressed as Taylor Series, and Convergence

Σx^n, converges (-1,1)

Arc Length of Parametric Curve

∫from a to b sqrt[x'(t)^2 + y'(t)^2 dt]

Area under a parametric Curve

A = ∫ from a to b f(t)+x'(t) dt

Telescoping Series

A series whose partial sums eventually only have a fixed number of terms after cancellation

Power Series

Cn(x-a)^n a = center cn = coefficients x = variable

Taylor's Remainder Theorem

Define the Remainder Rn(x) to be Rn(x) = f(x)−pn(x). Then Rn(x)=[(f(n+1)(c))/(n+1)!](x−a)n+1 for some 'c' between 'x' and 'a'. So |Rn(x)|≤[M/(n+1)!]|x−a|^(n+1)