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series-strategy2 - |< 1 a p-series converges if p> 1...

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Strategy for testing series We skipped Section 11.7, which is very short and contains some strategy for testing series. Here’s my version of that strategy. STEP 0. Does the n th term a n approach 0? If it doesn’t, then the series diverges. (I think of this as a “pre-test”.) In maybe 90% of problems, a n will approach 0. If a n approaches 0, you’re not done . The series might converge and might diverge – the problem is just beginning. We learned several tests for infinite series: basic comparison test (for series with positive terms) limit comparison test (for series with positive terms) integral test (for series with positive terms) alternating series test (for alternating series) ratio test We also learned about two “well-known” series: geometric series and p -series. Remember what those are , and remember when they converge . (A geometric series converges if | r |
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Unformatted text preview: | < 1; a p-series converges if p > 1.) The limit comparison test is more “versatile” than the basic comparison test. With either type of comparison test, you have to choose b n . In the limit comparison test, L = lim a n b n . If L 6 = 0 and L 6 = ∞ , the two series ∑ a n and ∑ b n behave the same. (It’s irrelevant whether L < 1 or L > 1.) The integral test is useful for series like ∑ 1 n ln n or ∑ 1 n (ln n ) 2 because the substitution u = ln x will help in the corresponding integral. The ratio test is useful for series containing factorials, and for power series. In the ratio test, L = lim | a n +1 a n | . If L < 1, the series converges absolutely (this includes L = 0). If L > 1, the series diverges (this includes L = ∞ ). The only value of L that gives you no information is L = 1. 1...
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