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Math 209 Final Exam – Concordia University 2026–2027 |
Comprehensive Practice Test, Advanced Calculus Concepts,
Step-by-Step Problem Solving, Detailed Solutions, Formula
Review, Exam Preparation Strategies, and Final Assessment
Study Guide
1. What is the McLaurin series representation for 1/(1x)?
A) ∑xⁿ/n!
B) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
C) ∑xⁿ
D) ∑(1)ⁿx²ⁿ/(2n)!
Correct Answer: C The McLaurin series for 1/(1x) is ∑xⁿ = 1+x+x²+x³+... with radius of
convergence R=1, representing a geometric series expansion.
2. Determine the radius of convergence for the McLaurin series of e^x:
A) R=0
B) R=1
C) R=∞
D) R=2
Correct Answer: C The exponential function's McLaurin series ∑xⁿ/n! converges for all real
numbers, giving an infinite radius of convergence R=∞, making it analytic everywhere.
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3. What is the McLaurin series expansion for sin(x)?
A) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
B) ∑xⁿ/n!
C) ∑(1)ⁿx²ⁿ/(2n)!
D) ∑x²ⁿ⁺¹/(2n+1)!
Correct Answer: A The sine function expands as x x³/3! + x⁵/5! x⁷/7! + ... with alternating
signs and odd powers, converging for all real x.
4. The McLaurin series for cos(x) has what general term?
A) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
B) ∑(1)ⁿx²ⁿ/(2n)!
C) ∑x²ⁿ/(2n)!
D) ∑(1)ⁿxⁿ/n!
Correct Answer: B Cosine expands as 1 x²/2! + x⁴/4! x⁶/6! + ... featuring even powers and
alternating signs with R=∞.
5. A series ∑aₙ is called absolutely convergent when:
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A) ∑aₙ converges
B) ∑|aₙ| converges
C) ∑aₙ diverges
D) ∑|aₙ| diverges
Correct Answer: B Absolute convergence means the series of absolute values converges,
which automatically implies the original series converges as well.
6. A conditionally convergent series is one that:
A) Converges absolutely
B) Diverges
C) Converges but not absolutely
D) Oscillates without converging
Correct Answer: C Conditional convergence occurs when the series itself converges, but the
series of absolute values diverges, such as the alternating harmonic series.
7. If ∑|aₙ| converges, what can we conclude about ∑aₙ?
A) It diverges
B) It converges absolutely
C) It converges conditionally
D) It may converge or diverge
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Correct Answer: B Absolute convergence of the absolute value series guarantees that the
original series converges absolutely, which is stronger than ordinary convergence.
8. According to the Root Test, if L = 1, what is the conclusion?
A) The series converges
B) The series diverges
C) The test is inconclusive
D) The series converges absolutely
Correct Answer: C When the Root Test yields L=1, no determination can be made about
convergence or divergence, requiring another test.
9. For the Ratio Test, if L < 1, the series:
A) Diverges
B) Converges absolutely
C) Converges conditionally
D) Is inconclusive
Correct Answer: B A ratio test result of L<1 indicates absolute convergence of the series,
meaning it converges regardless of sign variations.
Math 209 Final Exam – Concordia University 2026–2027 |
Comprehensive Practice Test, Advanced Calculus Concepts,
Step-by-Step Problem Solving, Detailed Solutions, Formula
Review, Exam Preparation Strategies, and Final Assessment
Study Guide
1. What is the McLaurin series representation for 1/(1x)?
A) ∑xⁿ/n!
B) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
C) ∑xⁿ
D) ∑(1)ⁿx²ⁿ/(2n)!
Correct Answer: C The McLaurin series for 1/(1x) is ∑xⁿ = 1+x+x²+x³+... with radius of
convergence R=1, representing a geometric series expansion.
2. Determine the radius of convergence for the McLaurin series of e^x:
A) R=0
B) R=1
C) R=∞
D) R=2
Correct Answer: C The exponential function's McLaurin series ∑xⁿ/n! converges for all real
numbers, giving an infinite radius of convergence R=∞, making it analytic everywhere.
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3. What is the McLaurin series expansion for sin(x)?
A) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
B) ∑xⁿ/n!
C) ∑(1)ⁿx²ⁿ/(2n)!
D) ∑x²ⁿ⁺¹/(2n+1)!
Correct Answer: A The sine function expands as x x³/3! + x⁵/5! x⁷/7! + ... with alternating
signs and odd powers, converging for all real x.
4. The McLaurin series for cos(x) has what general term?
A) ∑(1)ⁿx²ⁿ⁺¹/(2n+1)!
B) ∑(1)ⁿx²ⁿ/(2n)!
C) ∑x²ⁿ/(2n)!
D) ∑(1)ⁿxⁿ/n!
Correct Answer: B Cosine expands as 1 x²/2! + x⁴/4! x⁶/6! + ... featuring even powers and
alternating signs with R=∞.
5. A series ∑aₙ is called absolutely convergent when:
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A) ∑aₙ converges
B) ∑|aₙ| converges
C) ∑aₙ diverges
D) ∑|aₙ| diverges
Correct Answer: B Absolute convergence means the series of absolute values converges,
which automatically implies the original series converges as well.
6. A conditionally convergent series is one that:
A) Converges absolutely
B) Diverges
C) Converges but not absolutely
D) Oscillates without converging
Correct Answer: C Conditional convergence occurs when the series itself converges, but the
series of absolute values diverges, such as the alternating harmonic series.
7. If ∑|aₙ| converges, what can we conclude about ∑aₙ?
A) It diverges
B) It converges absolutely
C) It converges conditionally
D) It may converge or diverge
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Correct Answer: B Absolute convergence of the absolute value series guarantees that the
original series converges absolutely, which is stronger than ordinary convergence.
8. According to the Root Test, if L = 1, what is the conclusion?
A) The series converges
B) The series diverges
C) The test is inconclusive
D) The series converges absolutely
Correct Answer: C When the Root Test yields L=1, no determination can be made about
convergence or divergence, requiring another test.
9. For the Ratio Test, if L < 1, the series:
A) Diverges
B) Converges absolutely
C) Converges conditionally
D) Is inconclusive
Correct Answer: B A ratio test result of L<1 indicates absolute convergence of the series,
meaning it converges regardless of sign variations.
