← MATH 01131 · Calculus II
MATH 01131 · Sequences & series
Convergence tests (nth term, integral, comparison, ratio, root, alternating)
Definition
Toolbox for Σ aₙ: nth-term test (aₙ↛0 ⇒ diverge); integral; comparison/limit comparison; ratio/root; alternating series (Leibniz); absolute vs conditional convergence.
Read this
Always try nth-term first. Then match the series shape to a test (positive terms vs alternating).
Ratio/root shine on factorials and exponentials. Comparison needs a known p-series or geometric benchmark.
Absolute convergence (Σ|aₙ|) ⇒ convergence. Alternating harmonic converges conditionally.
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Try this
Attempt each problem first, then open the worked solution.
1.Test Σ 1/n², Σ (−1)ⁿ/n, Σ n!/nⁿ.