seriestests - and ∑ b n converges and ∑ b n diverges...

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A summary of the tests for series We will consider a series of the form n = a a n Name How to use it When to use it Does it tell you if Does it tell you if Example the series converges? the series diverges? n-th term test Calculate Always No Yes, if the limit a n = n n - 1 lim n →∞ a n is not equal to 0 Integral test Calculate When f ( x ) is easy Yes, if the Yes, if a n = 1 n ln n i a f ( x ) dx to integrate integral converges integral diverges Direct a n b n There is a cos n or sin n Yes, if it is less than Yes, if it is bigger than a n = sin 2 n n 2 Comparison or a n b n in the numerator a convergent series a divergent series Limit Calculate When a n is a fraction Yes, if L n = Yes, if L n = 0 a n = n +1 n 2 (1+ n ) comparison lim a n b n = L of n to various powers
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Unformatted text preview: and ∑ b n converges and ∑ b n diverges Ratio Calculate a n has z n Yes, if ρ < 1 Yes, if ρ > 1 a n = 3 n n ! , test lim n →∞ a n +1 a n = ρ and/or n ! in it power series Root Calculate a n has z n , or f ( n ) n Yes, if ρ < 1 Yes, if ρ > 1 a n = 2 n 3 n n n test lim n →∞ a 1 n n = ρ Also, n 1 n → 1 Alternating Show u n +1 ≤ u n Whenever you see Yes, if the three. No a n = (-1) n 3 n 2 n 3 +1 series test u n ≥ 0 and u n → ( − 1) n or ( − z ) n conditions are met. Check for absolute or conditional convergence. 1...
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This note was uploaded on 08/08/2008 for the course MATH 222 taught by Professor Wilson during the Fall '08 term at University of Wisconsin.

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