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Pos_hnd - (i If lim = ∞ → c b a n n n then either both...

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Positive-Term Series Theorems on Convergence or Divergence of a Positive-Term Series 1. If n a is a positive-term series and if there exists a number M such that M a a a S n n < + + + = 2 1 for every n , then the series converges and has a sum M S . If no such M exists, the series diverges. 2. Integral Test If n a is a series, let n a n f = ) ( and let f be the function obtained by replacing n with x . If f is positive-valued, continuous, and decreasing for every real number 1 x , then the series n a (i) converges if 1 ) ( dx x f converges. (ii) diverges if 1 ) ( dx x f diverges. 3 . P -series If p is a positive real number, then the p -series + + + + + = p p p p n n 1 3 1 2 1 1 1 1 (i) converges if 1 p (ii) diverges if 1 p 4. Basic Comparison Test Let n a and n b be positive-term series. (i) If n b converges and n n b a for every positive integer n , then n a converges. (ii) If n b diverges and n n b a for every positive integer n , then n a diverges. 5. Limit Comparison Test Let n a and n b be positive-term series.
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Unformatted text preview: (i) If lim = ∞ → c b a n n n then either both series converge or both series diverge. (ii) If lim = ∞ → n n n b a and ∑ n b converges, then ∑ n a also converges. (iii) If ∞ = ∞ → n n n b a lim and ∑ n b diverges, then ∑ n a also diverges. 6. Ratio Test Let ∑ n a be a positive-term series, and suppose that L a a n n n = + ∞ → 1 lim (i) If 1 < L , the series is convergent (ii) If ∞ = + ∞ → n n n a a L 1 lim or 1 , the series is divergent. (iii) If 1 = L , apply a different test; the series may be convergent or divergent. 7. Root Test Let ∑ n a be a positive-term series, and suppose that L a n n n = ∞ → lim (i) If 1 < L , the series is convergent (ii) If ∞ = ∞ → n n n a L lim or 1 , the series is divergent. (iii) If 1 = L , apply a different test; the series may be convergent or divergent....
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