Convergence tests, p-series, alternating series error, Taylor and Maclaurin series and radius of convergence.
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- State the Divergence Test for infinite series.
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If lim(n→∞) aₙ ≠ 0 or does not exist, then the series diverges.- What must be true about lim(n→∞) aₙ for a series to possibly converge?
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The limit must equal 0 (necessary but not sufficient condition).- What is a p-series?
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A series of the form sum(1/n^p) where p is a positive constant.- For what values of p does the p-series sum(1/n^p) converge?
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The series converges when p > 1.- The harmonic series sum(1/n) is an example of which p-series, and does it converge?
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It is the p-series with p = 1, and it diverges.- State the Integral Test condition for convergence.
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If f is positive, continuous, and decreasing on [1, ∞), then sum(aₙ) and integral(1 to ∞) f(x)dx converge together or diverge together.- What conclusion can you make if integral(1 to ∞) f(x)dx converges?
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The series sum(aₙ) where aₙ = f(n) also converges.- State the Limit Comparison Test.
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If aₙ > 0, bₙ > 0, and lim(n→∞) aₙ/bₙ = L where L is a positive finite number, then sum(aₙ) and sum(bₙ) converge together or diverge together.- State the Direct Comparison Test.
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If 0 ≤ aₙ ≤ bₙ for all n and sum(bₙ) converges, then sum(aₙ) converges. If sum(aₙ) diverges, then sum(bₙ) diverges.- State the Ratio Test for series convergence.
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Let L = lim(n→∞) |aₙ₊₁/aₙ|. If L < 1, series converges absolutely; if L > 1, series diverges; if L = 1, test is inconclusive.- State the Root Test for series convergence.
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Let L = lim(n→∞) ⁿ√|aₙ|. If L < 1, series converges absolutely; if L > 1, series diverges; if L = 1, test is inconclusive.- What is the sum of a convergent geometric series sum(arⁿ) where |r| < 1?
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The sum is a/(1 - r).- For what values of r does the geometric series sum(rⁿ) converge?
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The series converges when |r| < 1.- State the Alternating Series Test.
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If aₙ > 0, aₙ is decreasing, and lim(n→∞) aₙ = 0, then sum((-1)ⁿ aₙ) converges.- What is the Alternating Series Error Bound?
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If an alternating series satisfies the Alternating Series Test, the absolute error when using Sₙ to approximate the sum is at most |aₙ₊₁|.
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