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Unit 5: Differential Equations and Series

Study guide for CLEP CLEP Calculus — Unit 5: Differential Equations and Series. Practice questions, key concepts, and exam tips.

34

Practice Questions

20

Flashcards

4

Key Topics

Key Concepts to Study

separable DEs
slope fields
Taylor/Maclaurin series
convergence tests

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 34.

Q1HARD

Find the power series representation for the function f(x) = 1 / (1 + x) using the geometric series formula.

A) x + $x^{2}$ + $x^{3}$ + $x^{4}$ + ...
B) 1 + x + $x^{2}$ + $x^{3}$ + ...
C) x - $x^{2}$ + $x^{3}$ - $x^{4}$ + ...
D) 1 - x + $x^{2}$ - $x^{3}$ + ...
E) 1 + x - $x^{2}$ + $x^{3}$ - $x^{4}$ + ...
Show Answer

Answer: D — The geometric series formula is 1 / (1 - r) = 1 + r + $r^{2}$ + ... . For f(x) = 1 / (1 + x), let r = -x.

Q2HARD

Consider series: 1 - 2 + 3 - 4 + ... . What is its sum?

A) 1/2
B) 1/4
C) -1/2
D) 1
E) Does not exist
Show Answer

Answer: E — Does not exist is correct because series diverges by oscillation between 0 and 1

Q3MEDIUM

Determine if the series ∑[1/($n^{2}$)] from n=1 to ∞ converges.

A) Yes, by the ratio test
B) Yes, by the root test
C) Yes, by the p-series test
D) No, by the nth term test
E) No, by the integral test
Show Answer

Answer: C — Yes, by the p-series test is correct because it's a p-series with p > 1, so it converges.

Q4HARD

Determine whether the series ∑[n=1 to ∞] (1/n) converges or diverges.

A) The series converges by the ratio test.
B) The series converges by the alternating series test
C) The series converges by the root test.
D) The series converges by the limit comparison test with the series ∑[n=1 to ∞] (1/$n^{2}$).
E) The series diverges by the integral test.
Show Answer

Answer: E — The series is the harmonic series, which is known to diverge. The integral test can be applied to show this.

Q5EASY

Find sum of infinite geometric series with first term 2 and common ratio 1/2.

A) 2
B) 3
C) 4
D) 5
E) 6
Show Answer

Answer: C — 4 is correct because sum = a/(1-r) = 2/(1-1/2) = 4.

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Study Tips for Unit 5: Differential Equations and Series

  • Focus on understanding concepts, not memorizing facts — CLEP tests application
  • Practice with timed questions to build exam-day speed
  • Review explanations for wrong answers — they reveal common misconceptions
  • Use flashcards for key terms, practice questions for deeper understanding

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