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### Pos_hnd

Course: CALC 1,2,3, Spring 2008
School: Lehigh
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Word Count: 349

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Series Theorems Positive-Term on Convergence or Divergence of a Positive-Term Series 1. If a n is a positive-term series and if there exists a number M such that Sn a1 a2 an M for every n, then the series converges and has a sum S diverges. 2. Integral Test If M . If no such M exists, the series a n is a series, let f (n) a n and let f be the function obtained by replacing n with x. If f is positive-valued,...

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Series Theorems Positive-Term on Convergence or Divergence of a Positive-Term Series 1. If a n is a positive-term series and if there exists a number M such that Sn a1 a2 an M for every n, then the series converges and has a sum S diverges. 2. Integral Test If M . If no such M exists, the series a n is a series, let f (n) a n and let f be the function obtained by replacing n with x. If f is positive-valued, continuous, and decreasing for every real number x (i) converges if 1 1, then the series an f ( x)dx converges. f ( x)dx diverges. 1 (ii) diverges if 3. P-series If p is a positive real number, then the p-series (i) (ii) 1 1 1 p 2p 1 n converges if p 1 diverges if p 1 1 3p 1 np 4. Basic Comparison Test Let (i) (ii) a n and If If bn be positive-term series. bn converges and a n bn diverges and a n bn for every positive integer n, then bn every for positive integer n, then a n converges. a n diverges. 5. Limit Comparison Test Let (i) a n and If lim bn be positive-term series. (ii) (iii) an n bn an If lim n bn an If lim n bn c 0 then either both series converge or both series diverge. bn converges, then a n also converges. 0 and and bn diverges, then a n also diverges. 6. Ratio Test Let a n be a positive-term series, and suppose that lim (i) (ii) (iii) an 1 L n an If L 1 , the series is convergent a If L 1 or lim n 1 , the series is divergent. n an If L 1 , apply a different test; the series may be convergent or divergent. 7. Root Test Let a n be a positive-term series, and suppose that lim n a n n L (i) (ii) (iii) 1 , the series is convergent If L 1 or lim n a n , the series is divergent. n If L If L 1 , apply a different test; the series may be convergent or divergent.
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