Step by Step
1
Always start with the Divergence Test
Before trying anything else, check whether the limit of the series' individual terms, as n approaches infinity, is nonzero. If it is, the series automatically diverges, and no further testing is needed. If the limit IS zero, this test is inconclusive — you must try another test.
2
Geometric series
A geometric series converges if the absolute value of the common ratio r is less than 1, and in that case the sum equals a / (1−r), where a is the first term. If |r| ≥ 1, it diverges.
3
p-series
A p-series, of the form Σ 1/nᵖ, converges if p is greater than 1, and diverges if p is less than or equal to 1.
4
Ratio Test — best for factorials and exponentials
Compute the limit of |aₙ₊₁/aₙ| as n approaches infinity. If this limit is less than 1, the series converges; if greater than 1, it diverges; if exactly 1, the test is inconclusive. This test is especially effective for series involving factorials or exponential terms, since those ratios simplify nicely.
Applied Walkthrough
1
Determine whether Σ n!/(2ⁿ) converges. First, check the Divergence Test: the terms clearly grow without bound (factorials outgrow exponentials), so the limit of the terms isn't zero — but let's use the Ratio Test to confirm formally, since factorials are involved.
2
Since the series involves a factorial, the Ratio Test is the appropriate tool. Compute the ratio of consecutive terms: aₙ₊₁/aₙ = [(n+1)!/2ⁿ⁺¹] / [n!/2ⁿ] = (n+1)/2.
3
Take the limit as n approaches infinity: lim (n+1)/2 = ∞, which is greater than 1.
4
Since the Ratio Test limit is greater than 1, the series diverges — confirming what the Divergence Test also suggested.
Exam Application
Exams test whether you check the Divergence Test first before trying more complex tests, and whether you can correctly select and apply the appropriate test (geometric formula, p-series rule, or Ratio Test) based on the series' structure.
⚠ Common Trap
The most common trap is forgetting to check the Divergence Test first, or misinterpreting it — a limit of zero for the Divergence Test does NOT mean the series converges; it only means the test is inconclusive and another test is needed.
✓ Quick Self-Check
1. What should you always check first when testing a series for convergence?
The Divergence Test — whether the limit of the terms is nonzero.
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2. If the Divergence Test gives a limit of zero, what does that tell you?
Nothing conclusive — the test is inconclusive, and another test must be used.
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3. When does a geometric series converge, and what is its sum?
When |r| < 1; the sum is a / (1−r).
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4. When does a p-series converge?
When p > 1.
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5. Which test is best suited for series involving factorials or exponentials?
The Ratio Test.
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