Sequences & Series
Taylor & Maclaurin Series
Polar Area
Miscellaneous
Series Identification
100
What is the difference between a sequence and a series?
A sequence is a list of numbers while a series is a sum of numbers.
100

Find the first three terms of the Taylor series for x^3sin(x^3)

1+0x−(25/2)(x^2)

100
Find the slope of the curve r = 2 − 3sinθ at the point (2, π ).
2/3
100
What are parametric curves? Derivative of a Parametric Curve
Functions x and y defined by some third variable t dy/dx = (dy/dt)/(dx/dt)
100
∑1/n What type of series is this? Diverge or converge?
Harmonic series Diverges
200
What is the sum of the infinite geometric series 3/2 + 9/16 + 27/128 + 81/1024 +... ?
2.40
200
What is the coefficient of x^6 in the Taylor series expansion about x=0 for f(x)=sin(x^2)?
-1/6
200
What is the area of the region bounded inside the lemniscate r^2 = 4cos(2θ) and outside the circle r=√2?
2√3-2π/3
200
Position, Speed, and Acceleration of a Parametric Curve
Position: (x(t), y(t)) Speed: √[(dx/dt)²+(dy/dt)²] Acceleration: ((d²x/dt²),(d²y/dt²))
200
∑1/n^p What type of series is this? When does it converge/diverge?
p-Series converges when p>1, diverges when 0<p≤1
300
Determine whether the sequence (2n+3)/(5n-7) converges or diverges.
Converges to 2/5
300
Find the taylor series for f(x)= 1/(1+x)^2
1-2x+3x^2-4x^3+5x^4-...
300
Find the area enclosed by the curve. r=4+3sinθ
41π/2
300
Area of a Parametric Curve
A= α
300
MacLaurin Series: e^x
∑xⁿ/n! = 1+x+x²/2! + x³/3!...
400
Determine whether the sequence (2^n +3)/(2^n +1)is increasing, decreasing, or not monotonic.
Decreasing
400
Find the interval of convergence for the power series of -ln(1-x) centered at x=1/2.
0<x<1
400
Find the area inside one loop of r=sin^2(θ).
3π/16
400
Arc Length of a Parametric Curve
L = α
400
MacLaurin Series: sinx
∑(-1)ⁿx²ⁿ⁺¹ / (2n+1)! = x-x³/3!+x⁵/5!-x⁷/7!...
500
Discuss the convergence of the following series whose nth term is given: 2^(n-1)/3^(n+1).
Converges by the limit comparison test
500
Find the interval of convergence of the Maclaurin series of sin(x^2).
-∞<x<∞
500
Find the area inside one loop of r=cos(3θ).
π/12
500
Derivative of a Parametric Curve
dy/dx = (dy/dt)/(dx/dt)
500
MacLaurin Series: cosx
∑(-1)²ⁿx²ⁿ / (2n)! = 1-x²/2!+x⁴/4!-x⁶/6!...
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