Lesson 20a - Comparison Test

Youshouldnotethat

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Unformatted text preview: lid. This means since we know out left series diverges (P‐Test) we know that our right series must also diverge as it is larger than the known divergent series. Strategy: Comparison Test This means the strategy we should follow for using the comparison test is: Step 1: Determine the largest growing term in the numerator with the denominator. This will give you a clue to prove divergence or convergence. (you should note that ln(n) terms are smaller than n0.0001, which may make comparisons easier to do). Step 2: Show that the new series you are comparing to is smaller (if you are proving divergence getsmaller from your original) or larger (if you are proving convergence get larger from your original). This is proving the test will diverge/converge by the comparison test. Note: you may need to perform one of the sneaky tricks below to do this: To make the term bigger you can: Get rid of negative terms in the numerator Get rid of positive terms in the denominator Change positive terms in the numerator to different larger positive terms (change n to n2) Change negative terms in the denominator to different larger negative terms (change n to n2) Example 1 Determine if the following series converges or diverges: 2n 2 n 1 n 3 5n 2 2 n 1 First we determine our guess by comparing the largest terms: 2n 2 n 1 2n 2 1 3 2 n3 5n 2 2 n1 n n 1 n 1 n The latter series diverges (P‐test, p=1≤1) so we will want to show that the latter series (or some form of it) will be smaller than the original. We want to make our original larger, so we can...
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