Question: Suppose ∑ aₙ and ∑ bₙ are series with positive terms and ∑ bₙ is convergent. What can you say about ∑ aₙ?
Answer Choices:
If lim (aₙ/bₙ) = ∞, then ∑ aₙ is divergent.
If lim (aₙ/bₙ) = 0, then ∑ aₙ is convergent.
If lim (aₙ/bₙ) = 1, then ∑ aₙ is convergent.
None of the above.
Answer: First and Third options
Question: Convert (√3, –1) to polar coordinates with r ≥ 0 and θ between 0° and 360°.
Answer Choices:
(Numerical fill-in for r, θ)
Answer: (2, 330°)
Question: Suppose a₁ = 2 and aₙ₊₁ = (2/n) aₙ. Then the series ∑ aₙ …
Answer Choices:
Converges by Ratio Test …
Diverges by Ratio Test …
Converges by Root Test …
Diverges by Root Test …
Answer: Converges by Ratio Test, L = 0
Question: Given r = 2 – 2cos(θ), determine the following:
A) r has a horizontal tangent at θ = 2π/3.
B) r has a vertical tangent at θ = 2π/3.
Answer Choices:
True / False
Answer: A) False, B) True
Question: Apply the nᵗʰ Root Test to the series ∑ [n·tan⁻¹(n)]^(n^½).
Answer Choices:
The series converges by Root Test because L = …
The series diverges by Root Test because L = …
The series diverges by Root Test because L → ∞.
The series cannot be applied because L = …
Answer: Diverges by Root Test because L → ∞
Question: The equation r² = 2cos(2θ) in polar form is equivalent to which equation in rectangular form?
Answer Choices:
x² + y² = yx
(x² + y²)² = 2x² – 2y²
x² + y² = 6yx
(x + y)² = 6yx
Answer: (x² + y²)² = 2x² – 2y²
Question: Suppose ∑ aₙ and ∑ bₙ are series with positive terms and ∑ bₙ is divergent. What can you say about ∑ aₙ?
Answer Choices:
If lim (aₙ/bₙ) = 0, then ∑ aₙ is divergent.
If lim (aₙ/bₙ) = ∞, then ∑ aₙ is convergent.
If lim (aₙ/bₙ) = 1, then ∑ aₙ is convergent.
None of the above.
Answer: None of the above
Question: The equation r = 8sin(θ) in polar form is equivalent to which equation in rectangular form?
Answer Choices:
x² + 8y² = 0
x + y = 8
x² – 8y² = 0
x² + y² – 8y = 0
Answer: x² + y² – 8y = 0
Question: The series ∑ [(-3)^(n+1) / 2^(2n)] … (choose all correct).
Answer Choices:
Is geometric and converges as |r| < 1
Converges by Ratio Test
Converges by Root Test
Converges by Alternating Series Test
Is absolutely convergent
None of the above
Answer: Is geometric and converges, Converges by Ratio Test, Converges by Root Test, Is absolutely convergent
Question: Given r = 8 – 8sin(θ), find the following:
A) The maximum radius.
B) The graph symmetry.
Answer Choices:
(Numeric + multiple choice symmetry line)
Answer: A) 16, B) symmetric about line θ = π/2
Question: The series ∑ [nⁿ / (ln(n))^(2n)] …
Answer Choices:
Converges by Ratio Test …
Converges by Root Test …
Diverges by Ratio Test …
Diverges by Root Test …
Answer: Diverges by Root Test, L = ∞