Series convergence, uniform convergence, mean value theorem, Riemann integrability and function sequences.
39 cards · basic cards · AI-written, checked twice. Edit anything.
- Define convergence of a series sum_{n=1}^∞ a_n.
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The series converges if the sequence of partial sums S_N = sum_{n=1}^N a_n has a finite limit as N approaches infinity.- What is the necessary condition on a_n for sum a_n to converge?
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lim_{n→∞} a_n = 0.- For what values of r does the geometric series sum_{n=0}^∞ r^n converge, and to what?
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Converges when |r| < 1, with sum 1/(1-r).- Does the harmonic series sum_{n=1}^∞ 1/n converge or diverge?
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Diverges.- State the Ratio Test for series sum a_n.
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If lim_{n→∞} |a_{n+1}/a_n| = L, the series converges absolutely if L < 1, diverges if L > 1, and is inconclusive if L = 1.- State the Root Test for series sum a_n.
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If lim_{n→∞} (|a_n|)^(1/n) = L, the series converges absolutely if L < 1, diverges if L > 1, and is inconclusive if L = 1.- State the Comparison Test.
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If 0 ≤ a_n ≤ b_n for all n: if sum b_n converges then sum a_n converges; if sum a_n diverges then sum b_n diverges.- State the Limit Comparison Test.
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If a_n, b_n > 0 and lim_{n→∞} a_n/b_n = L with 0 < L < ∞, then sum a_n and sum b_n either both converge or both diverge.- State the Integral Test.
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If f is positive, decreasing, and continuous, then sum_{n=1}^∞ f(n) and integral_1^∞ f(x)dx either both converge or both diverge.- State the Alternating Series Test.
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If a_n > 0, {a_n} is decreasing, and lim_{n→∞} a_n = 0, then sum_{n=1}^∞ (-1)^n a_n converges.- Define absolute convergence of a series.
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sum a_n converges absolutely if sum |a_n| converges.- For which values of p does the p-series sum_{n=1}^∞ 1/n^p converge?
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Converges if and only if p > 1.- State the Cauchy Criterion for convergence of a series.
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sum a_n converges if and only if for every ε > 0 there exists N such that |sum_{n=N+1}^M a_n| < ε for all M > N.- Define pointwise convergence of a function sequence {f_n} to f on a set S.
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For each x in S, lim_{n→∞} f_n(x) = f(x).- Define uniform convergence of a function sequence {f_n} to f on a set S.
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For every ε > 0, there exists N such that |f_n(x) - f(x)| < ε for all x in S and all n > N.
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