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ConvTests - CONVERGENCE TESTS(1 Geometric series initial...

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CONVERGENCE TESTS (1) Geometric series: initial term = a, common ratio = r a + ar + ar 2 + . . . + = n = 1 a r n - 1 , n = 0 a r n . (i) Converges when - 1 < r < 1. (ii) Diverges when r 1 or r ≤ - 1. If convergent a + ar + ar 2 + . . . + = a 1 - r . (2) Divergence Test: lim n → ∞ a n = 0 = n a n diverges (3) Integral test: If f is positive, continuous and decreasing on [1 , ), then 1 f ( x ) dx converges ⇐⇒ n = 1 f ( n ) converges (4) Special Cases: n = 1 1 n p converges p > 1 , diverges p 1 , n = 1 1 n (ln n ) q converges q > 1 , diverges q 1 , (5) Comparison Test: If 0 a n b n , then n = 1 b n converges = n = 1 a n converges . If 0 b n a n , then n = 1 b n diverges = n = 1 a n diverges . (6) Limit Comparison Test: If 0 a n , b n and lim n → ∞ a n b n = L, L = 0 , then n = 1 b n converges = n = 1 a n converges n = 1 b n diverges = n = 1 a n diverges (7) Alternating Series Test: If b n > 0, then b 1 - b 2 + b 3 - b 4 + . . . = n 1 ( - 1) n - 1 b n converges if (i) b n b n +1 , (ii) lim n → ∞ b n = 0 A series a n is said to CONVERGE ABSO- LUTELY if the series | a n | of absolute values converges. Notice (i) if a n 0, then convergence of a n is the same as the Absolute Convergence of a n ; (ii) if a n converges absolutely, then
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