final prep

Ratio Test - convergent

Lim (n -> infinity) | a | < 1

Ratio Test - divergent

Lim (n -> infinity) | a | > 1

Ratio Test - inconclusive

Lim (n -> infinity) | a | = 1

Sum of Geometric Series

a / 1 - r

Alternating Series Test - Convergent

lim (n -> infinity) | a | = 0

Alternating Series Test - Divergent

lim (n -> infinity) | a | is not 0

Geometric Series - Convergent

| r | < 1

Geometric Series - Divergent

| r | >= 1

Test for Divergence - Divergent

lim (n -> infinity) is not 0

Test for Divergence - Inconclusive

lim (n -> infinity) = 0

Integral Test - Convergent

Integral of a = finite number

Integral Test - Divergent

Integral of a = (-) infinity

P-Series - Convergent

p > 1

P-Series - Divergent

p <= 1

Harmonic Series

p = 1 (always divergent)

Direct Comparison Test - Convergent

Compared series is larger and convergent

Direct Comparison Test - Divergent

Compared series is smaller and divergent

Direct Comparison Test - Inconclusive

Compared series is larger and divergent or smaller and convergent

Limit Comparison Test - Convergent

Lim (n -> infinity) series/compared series = positive, finite number AND compared series is convergent

Limit Comparison Test - Divergent

Lim (n -> infinity) series/compared series = positive, finite number AND compared series is divergent

Limit Comparison Test - Inconclusive

Lim (n -> infinity) series/compared series = infinity or 0

Root Test - Convergent

lim (n -> infinity) | a | < 1

Root Test - Divergent

lim (n -> infinity) | a | > 1

Root Test - Inconclusive

lim (n -> infinity) | a | = 1

Absolute Convergence

| a | converges and a converges

Conditional Convergence

| a | diverges and a converges

Divergence

| a | diverges and a diverges

Maclaurin series - sin x

(Sigma - infinity, n = 0) x^2n+1 / (2n+1)! * (-1)^n

Maclaurin series - e^x

(Sigma - infinity, n = 0) x^n / n!

Maclaurin series - cos x

(Sigma - infinity, n = 0) x^2n / (2n)! * (-1)^n

Maclaurin series - arctan x

(Sigma - infinity, n = 0) x^2n+1 / 2n+1 * (-1)^n

Pythagorean Identity (sin/cos)

sin^2 x + cos^2 x = 1

Pythagorean Identity (tan/sec)

tan^2 x = sec^2 x - 1

Pythagorean Identity (cot/csc)

cot^2 x = csc^2 x - 1

Integration by Parts

Integral of u dv = uv - integral of v du

Trig Substitution of a^2 - x^2

x = a sin theta

Trig Substitution of x^2 - a^2

x = a sec theta

Trig Substitution of x^2 + a^2

x = a tan theta

Half angle Identity (sin^2 x)

(1 - cos (2x)) / 2

Half angle Identity (cos^2 x)

(1 + cos (2x)) / 2

Integral of csc x

ln | cscx - cotx| + c

Integral of tan x

ln | secx | + c

Integral of cot x

ln | sinx | + c

Integral of sec x

ln | secx + tanx | + c

Integral of sec^3 x

(secx * tanx + ln | secx + tanx |) / 2 + c

Midpoint Approximation

delta x ( f( (x0 + x1) / 2 ) + f( (x1 + x2) / 2 ) + . . . )

Trapezoid Approximation

delta x / 2 ( f(x0) + 2f(x1) + 2f(x2) + . . . + f(xn) )

Simpson's Rule Approximation

delta x / 3 ( f(x0) + 4f(x1) + 2f(x2) + . . . + 4f(xn-1) + f(xn) )

Partial Fractions of x(x^2 + 1)

A / x + (Bx + C) / (x^2 + 1)

Partial Fractions of x(x + 1)

A / x + B / (x + 1)

Partial Fractions of x^2

A / x + B / x^2

Partial Fractions of (x + 1)(x + 2)

A / (x + 1) + B / (x + 2)

Taylor Series

f^(n) (a) / n! * (x - a)^n

Work

w = F*d

Spring formula

F(x) = kx

Force

F = m*a

Cartesian to Polar - x

x = r cos theta

Cartesian to Polar - y

y = r sin theta

Polar to Cartesian - r

r^2 = x^2 + y^2

Polar to Cartesian - theta

theta = tan^-1 (y/x)

Parametric dy/dx

dy/dx = dy/dt / dx/dt

Parametric second derivative

d^2 y/d x^2 = d/dt (dy/dx) / dx/dt

Polar dy/dx

dy/dx = (r' sin theta + r cos theta) / (r' cos theta - r sin theta)

Arc Length - Cartesian

integral (a -> b) of sqrt ( 1 + (f'(x))^2 )

Arc Length - Parametric

integral (a -> b) of sqrt ( (dx/dt)^2 + (dy/dt)^2 )

Arc Length - Polar

integral (a -> b) of sqrt ( r^2 + (dr/dtheta)^2 )

Area - Polar

1/2 integral (a -> b) r^2

integration by parts prioritization

LIPET -

L - Logarithmic (e.g., \ln x, \log x)

I - Inverse trigonometric (e.g., \tan^{-1}x, \sin^{-1}x)

P - Polynomial (e.g., x^2, 3x)

E - Exponential (e.g., e^x, 2^x)