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Unformatted text preview: j k where k is a positive integer, then both series converge or both series diverge. VI. For any positive integer k , the series a a a n n = + + ⋅⋅⋅ = ∞ ∑ 1 2 1 and a a a n k k n = + + ⋅⋅⋅ + + = ∞ ∑ 1 2 1 either both converge or both diverge. VII. If a n ∑ and b n ∑ are convergent series with sums A and B , respectively, then (i) ( ) a b n n + ∑ converges and has sum A B + (ii) ca n ∑ converges and has sum cA for every real number c (iii) ( ) a b n n-∑ converges and has sum A B-VIII. If a n ∑ is a convergent series and b n ∑ is divergent, then the series ( ) a b n n + ∑ is divergent....
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