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Tests For Convergence I. Return Midterm II II. Homework III. Definition of Convergence Let an be a sequence then we define the nth partial sum of an as sn = a1 + a2 + ... + an In other words, we define sn by adding up the first n terms of an We define the series as the limit of the sn that is S = sum from i = 1 to infinity of an = a1 + a2 + a3 + ... If the limit exists then we say that the series converges. Otherwise, we say that the series diverges. IV. Geometric Series We define a geometric series to be a series of the form sum arn For example: 3/2 + 3/4 + 3/8 + ... Theorem: For 0 < |r| < 1 we have sum arn = a/(1 - r) and for |r| > 1 the series diverges. V. The Limit Test Theorem: If sum an converges then the limit as n -> infinity of an = 0 Note: The contrapositive says that if the limit is nonzero, then the series does not converge. Caution: if the limit goes to zero then the series still may diverge. Examples A) sum (n + 3)/(n + 2) diverges by the limit test since the limit is 1 not 0. B) sum 1/n does not converge even though the limit goes to 0. VI. The Ratio Test Theorem: The Ratio Test Let sum an be a series then 1) If lim |an+1/an| < 1 then the series converges absolutely 2) If |an+1/an| >1 then the series diverges 3) |an+1/an| = 1 then try another test. Example Determine the convergence or divergence of sum 4n/n! We use the Ratio Test: lim 4n+1/(n+1)!/4n/n! = lim [n!/(n+1)!][4n+1/4n = lim 4/(n + 1) = 0 Hence the series converges by the Ratio Test Exercises Determine the convergence or divergence of A) sum n!/[n(3n)] B) sum 2n/3n+1 VII. P-Series Theorem: P-Series Test Consider the series sum 1/np If p > 1 then the series converges If 0 < p < 1 then the series diverges Example: Show that the sum 1/n diverges. Solution: Since 1/n = 1/n1 and 1 > 1, the P-Series tests tells us that the series diverges Error The error of using Nterms to approximate a convergent P-Series is less that 1/(p-1)Np-1 Exercise: Approximate sum 1/n2 accurate to 4 decimal places.
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