MATH-152 EXAM III TAMU

Converges by nth term test when

N/A

Diverges by nth term test when

lim an ≠ 0
n→∞

Converges by Geometric Series Test when

|r| < 1

Diverges by Geometric Series Test when

|r| ≥ 1

Converges by p-series test when

p > 1

Diverges by p-series test when

p ≤ 1

Converges (absolutely) by Ratio test when

lim |an+1 / an| = L < 1
n→∞

Diverges by Ratio test when

lim |an+1 / an| = L > 1
n→∞

Converges by Direct Comparison test when

an ≤ bn, and ∑bn converges

Diverges by direct comparison test when

bn ≤ an, and ∑bn diverges

Converges by limit comparison when

lim(an / bn) > 0, is finite & bn converges
n→∞

Diverges by limit comparison when

lim(an / bn) > 0, is finite & bn diverges
n→∞

Converges by Alternating series test when

lim an = 0 & an+1 ≤ an for all n
n→∞

Diverges by Alternating series test when

N/A

Converges by integral test when

∫conv. => ∑conv.

Diverges by integral test when

∫div. => ∑div.

Sum of a geometric series

S = (a₁ / 1−r)

Converges (absolutely) by root test when

lim n√|an| = L < 1
n→∞

Diverges by root test when

lim n√|an| = L > 1
n→∞

Remainder Estimate for the Integral Test

Rn ≤ ∫an

Remainder Estimate for the Comparison Test

Rn ≤ Tn, where Tn is remainder from bn

Alternating Series Estimation Theorem

|Rn| = |s-sn| ≤ bn+1

Taylor's Inequality

|Rn(x)| ≤ M / (n+1)! |x-a|^(n+1)

Half Angle sin^2(x)

(1/2)(1-cos(2x))

Half Angle cos^2(x)

(1/2)(1+cos(2x))

d/dx arcsin(x)

1/√(1-x^2)

d/dx arccos(x)

-1/√(1-x^2)

d/dx arctan(x)

1/(1+x^2)

d/dx arccot(x)

-1/(1+x^2)

d/dx arcsec(x)

1/(x*√(x^2-1))

d/dx arccsc(x)

1/(x*√(x^2-1))

If the power of cosine is odd

save one cosine factor

If the power of sine is odd

save one sine factor

If the powers of both sine and cosine are even

use the half-angle identities

If the power of secant is even

save a factor of sec^2(x)

If the power of tangent is odd

save a factor of sec(x)tan(x)

∫tan(x)

ln|sec(x)| + C

∫sec(x)

ln|sec(x)+tan(x)| + C

Trig Sub for √(a^2 - x^2)

x = asin(θ)

Trig Sub for √(a^2 + x^2)

x = atan(θ)

Trig Sub for √(x^2 - a^2)

x = asec(θ)

sin(x)cos(x)

(1/2)sin(2x)

Area of parametric equation

∫g(t)f'(t)*(d/dt) when y = g(t) and x = f(t)

Tangent of parametric curve

dy/dx = (dy/dt)/(dx/dt)

Second derivative of parametric curve

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

Length of parametric curve

L = ∫√((dx/dt)^2 + (dy/dt)^2)

Surface Area of parametric curve when rotated around x axis

S = ∫(2πy√((dx/dt)^2 + (dy/dt)^2))dt

Surface Area of parametric curve when rotated around y axis

S = ∫2πx√((dx/dt)^2 + (dy/dt)^2)

Polar coordinates to rectangular coordinates

x = rcos(θ), y = rsin(θ)

Rectangular coordinates to polar coordinates

r = √(x^2 + y^2), θ = arctan(y/x)