Alternating Series Estimation Theorem: If Sₙ=Σⁿᵢ₌₁(-1)ⁿbₙ or Σⁿᵢ₌₁(-1)ⁿ⁻¹bₙ is the sum of an alternating series that converges, then |Rₙ|=|S-Sₙ|≤bₙ₊₁
Note that a Harmonic Series is equal to a P-Series with p = 1, which diverges according to the P-Series test (diverges if p≤1)
Nth root of |aₙ| = |aₙ|1/n
If Test for Divergence passes (lim n->∞=0), use another test
If a test is inconclusive, use another test
For the series listed, assume each sum to be an infinite sequence: Σ∞ₙ₌₀ = Σ
Note that the formula for a Degree 1 Taylor Polynomial has the same exact formula as the one used for Linear Approximation learned in Calculus I.
f(n) means "the nth derivative of the function f"
n! = n(n-1)! = n(n-1)(n-2)! = n(n-1)(n-2)(n-3)! = ...
n! = n(n-1)(n-2)(n-3)...*3*2*1
0! = 1, 1! = 1
Areas of Functions
Area of polar functions is with respect to the origin as the pole.
Average value of a function = 1/(b-a)∫ₐᵇ f(x)dx
Integration by Parts
The order of choosing your "u" variable goes by LIPET: Logarithms, Inverse trigonometry, Polynomials, Exponential functions, then Trigonometry.
Integration by Parts is used to integrate integrals that have components multiplied together in their simplest form. It is often referred to as a "product rule for integrals."
With respect to the x-axis: use dx
With respect to the y-axis: use dy
For Cylindrical Shells: radius = x or y, and height = f(x) or f(y)
Integration by Partial Fractions
Integration by Partial Fractions is used to simplify integrals of polynomial rational expressions into simpler fractions. It can only be used when the degree (highest power) of the numerator is less than the degree of the denominator. If the degree of the numerator polynomial is equal to or higher than the degree of the denominator, you must use polynomial long division before converting to partial fractions to integrate.
Parametric Curves and Polar Functions
(dx/dt) ≠ 0, (dx/dθ) ≠ 0,
For standard functions: f'(x) = dy/dx
For parametric functions: x'(t) = dx/dt and y(t)' = dy/dt
For polar functions: r'(θ) = dr/dθ
Simpson's Rule can only be used if the given n is even
In order of most accurate to least accurate: Simpson's Rule, Midpoint Rule, Trapezoidal Rule
Integral Approximations are typically used to evaluate an integral that is very difficult or impossible to integrate.
f'(x)=dy/dx, g'(y)=dx/dy, x'(t)=dx/dt, and y'(t)=dy/dt
The symbols "_∞" and "_-∞" represent infinity and negative infinity as a lower bound of an integral.