Determining Convergence - Determining Convergence for First use the nth-Term Test Is lim an = 0 If no then the series diverges If yes or maybe then use

Determining Convergence - Determining Convergence for First...

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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 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 ρ ρ ρ + →∞ - = < = VII. The nth-Root Test: ( 1) , lim 1) 1 2)13)1Series diverges ifpor=VIII.The Alternating Series:1123411*( 1)......:1)'2)3)nnnnnnnForaaaaaconverges if all three are satisfiedThe as are all positiveaafor all naas n+=+-=-+-+→ ∞ n n n n n For a a Series converges if Inconclusive if ρ ρ ρ →∞ - = <