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Geometric Series Test The series ∑(1,∞)arⁿ⁻¹ = a + ar + ar² + ... is convergent if |r|<1 and its sum is ∑(1,∞)arⁿ⁻¹ = a/(1-r) If |r|≥1, the series is divergent Divergence Test If lim as n->∞ ≠ 0, then ∑a(n) diverges Integral Test Given a sequence a(n), if f(n) = a(n) is a positive, decreasing function for x∈R, then i) ∫(1,∞) f(x)dx converges -> ∑(1,∞) a(n) converges ii) ∫(1,∞) f(x)dx diverges -> ∑(1,∞) a(n) diverges P-Rule ∑ from n=1 to n=∞ (1/n^p) converges IFF p>1 Comparison Test Given ∑a(n), if you can find a series ∑b(n) S.T. 1) ∑b(n) converges and a(n)≤b(n), then ∑a(n) converges 2) ∑b(n) diverges and a(n)≥b(n), then ∑a(n) converges Limit Convergence Test Given ∑a(n), find b(n) S.T. If lim n->∞ a(n)/b(n) = C where C≠0 then ∑a(n) & ∑b(n) either both converge or diverge If lim n->∞ a(n)/b(n) = 0 or ∞ then test fails Alternating Series Test Given ∑a(n) = ∑ (-1)ⁿb(n) If 1) lim n->∞ b(n) = 0 2) b(n) is a positive, decreasing sequence then ∑a(n) = ∑ (-1)ⁿb(n) converges Ratio Test Given ∑a(n) If 1) lim n->∞ |(a(n+1)/a(n)| < 1 then ∑a(n) absolutely converges 2) lim n->∞ |(a(n+1)/a(n)| > 1 then ∑a(n) diverges 3) lim n->∞ |(a(n+1)/a(n)| = 1 then test fails Roots Test Given ∑a(n) If 1) lim n->∞ n√(|a(n)|) < 1 then ∑a(n) converges 2) lim n->∞ n√(|a(n)|) > 1 then ∑a(n) diverges 3) lim n->∞ n√(|a(n)|) = 1 then test fails Monotonic Sequence Test If a sequence a(n) is bounded and only decreasing or only increasing, then a(n) has a limit