Determining Convergence

Determining Convergence - b a If and b diverges then a...

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Determining Convergence for n a First use the nth-Term Test : Is lim a n = 0? If no , then the series diverges. If yes or maybe, then use one of the tests below: I. Geometric Series Test : 2 * ..... /(1 ) 1 1 n For a a ar ar Converges to a r if r Diverges if r = + + + - < II. P-Series Test : 1 1 * , 1 1 p n For series converges if p n series diverges if p = III. Integral Test : { } * ( ) n n c c n a and f x dx both converge or both diverge if a has only positiveterms = IV. Direct Comparison Test for Series of Nonnegative Terms : 1) 2) n n n n n n n n a converges if there is a convergent series c with a c a diverges if there is a divergent series d with a d V. Limit Comparison Test for Series of Nonnegative Terms : 1) lim 0, 2) lim 0 , 3) lim , n n n n n n n n n n n n n n n a and b both converge or diverge a If c then b a If and b converges then a converges
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Unformatted text preview: b a If and b diverges then a diverges b →∞ →∞ →∞ = = = ∞ ∑ ∑ ∑ ∑ ∑ ∑ VI. Ratio Test : 1 * ( 1) , lim 1) 1 2) 1 3) 1 n n n n n a For a a Series converges if Series diverges if or Inconclusive if ρ + →∞-= < ∞ = ∑ VII. The nth-Root Test: ( 1) , lim 1) 1 2) 1 3) 1 n n n n n For a a Series converges if Series diverges if p or Inconclusive if →∞-= < ∞ = ∑ VIII. The Alternating Series: 1 1 2 3 4 1 1 * ( 1) ...... : 1) ' 2) 3) n n n n n n n For a a a a a converges if all three are satisfied The a s are all positive a a for all n a as n ∞ + = +-=-+-+ ≥ → → ∞ ∑...
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This note was uploaded on 03/22/2009 for the course MATH M112 taught by Professor Levy during the Fall '05 term at NJIT.

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Determining Convergence - b a If and b diverges then a...

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