Series Convergence Quiz

Grade 12 · Math · Quiz

10 questions applying convergence tests to infinite series.

Questions

Does Σ 1/n (harmonic series) converge?
1. Yes, to ln 2
2. Yes, to 1
3. No, it diverges
4. Cannot determine
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Answer: No, it diverges
The harmonic series is the p-series with p=1. Since p ≤ 1, it diverges.
The geometric series Σ 2·(3/4)ⁿ (n=0 to ∞) converges to:
1. 4
2. 6
3. 8
4. 2
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Answer: 8
a=2, r=3/4. Sum = 2/(1-3/4) = 2/(1/4) = 8.
Σ 1/n^(1/2) converges.
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p = 1/2 ≤ 1, so this p-series diverges.
Apply the Ratio Test to Σ 2ⁿ/n!. Does it converge or diverge?
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L = lim|2ⁿ⁺¹/(n+1)! · n!/2ⁿ| = lim 2/(n+1) = 0 < 1. Converges.
Which test is best for Σ 1/(n² + 3n)?
1. Geometric test
2. Ratio test
3. Comparison or limit comparison
4. nth-term test
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Answer: Comparison or limit comparison
Limit comparison with Σ1/n² (p=2, converges) shows this series also converges.
If the Ratio Test gives L = 1, the series definitely diverges.
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L = 1 is inconclusive — the series may converge or diverge.
Σ (-1)ⁿ diverges because:
1. r > 1
2. lim aₙ ≠ 0
3. p ≤ 1
4. Ratio test fails
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Answer: lim aₙ ≠ 0
lim (-1)ⁿ does not equal 0 (it oscillates between -1 and 1), so the series diverges by the nth-term test.
What is the sum of the geometric series Σ (1/2)ⁿ for n=1 to ∞?
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a = 1/2, r = 1/2. Sum = (1/2)/(1-1/2) = (1/2)/(1/2) = 1.
For what values of x does Σ xⁿ (geometric series, n=0 to ∞) converge?
1. |x| < 1
2. x < 1
3. x > 0
4. x ≠ 0
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Answer: |x| < 1
A geometric series converges when |r| < 1, so here when |x| < 1.
Every absolutely convergent series is also convergent.
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Absolute convergence (Σ|aₙ| converges) implies convergence of Σaₙ.

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