Limit comparison test suppose that k 1a k and k 1b k

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Limit Comparison Test Suppose that k =1 a k and k =1 b k are series with (eventually) all positive terms. (a) If lim k →∞ a k b k = C where 0 < C < then either both series converge or both diverge. (b) If lim k →∞ a k b k = and k =1 b k diverges then so does k =1 a k . (c) If lim k →∞ a k b k = 0 and k =1 b k converges then so does k =1 a k (or if k =1 a k diverges then so does k =1 b k ). Alternating Series Test (A.S.T.) If the alternating series k =1 ( - 1) k a k (where a k 0 for all k ) satisfies the two conditions (a) lim k →∞ a k = 0 (b) a k +1 a k for all k then the series converges. Alternating Series Estimation Theorem Let S denote the sum of a convergent alternating series (satisfying A.S.T). If we use the partial sum S n to approximate S then the error satisfies | S - S n | ≤ a n +1 . Ratio Test Suppose lim k →∞ a k +1 a k = L . Then (a) If L < 1 then the series k =1 a k converges absolutely. (b) If L > 1 then the series k =1 a k diverges. Also if L = , the series diverges. (c) If L = 1 then the test gives no information. Root Test Same as ratio test but with L = lim k →∞ | a k | 1 k = lim k →∞ k p | a k | .

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