Lesson 20a - Comparison Test


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Unformatted text preview: rges or diverges n n 1 2 5 To determine if the series converges or diverges using the comparison test, we need to have an idea if the series converges or diverges overall (you need to choose the right path). To do so, we determine the “largest growing term” as n tends to infinity in both the numerator and denominator. Here is a typical hierarchy that you should become familiar with: nn > n! > bn (b>1) > nb > ln(n) > c One notable thing is that ln(n)b is always (eventually) smaller than no matter what b is n0.0000000001 (b can be extremely large, and the miniscule polynomial will still win eventually). 1 n3 n n 2 5 n1 n n1 n This means we should try comparing to 2 n 1 We know that the latter series diverges (P‐Test p=1≤1) so we will work towards divergence. Strategy: Comparison Test n3 n 1 Since we want to compare to the divergent series , we n2 5 n2 n n 1 n 1 n 1 want to make sure: n3 1 n n n2 n2 5 n 1 n 1 n 1 When we take away positives from the numerator our final result will become smaller. When we add positives (take away negatives) from the denominator, our final result will still become smaller. This means the inequality presented above is va...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.

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