theorems - n a n = 0 3. Section 11.3 (a) The Integral Test:...

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Theorems you should know how to prove. *DISCLAIMER* The theorems listed below are NOT stated in their entirety. These are the parts of the theorems that you need to know how to prove. For example, the Geometric Series as stated below is incomplete since it does not mention the case where | r | ≥ 1 and so you are not responsible for proving that part. You still have to know ALL the complete statements of the theorems for the exam, but you only have to know how to prove the partial results below. 1. Section 11.1 (a) Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent. 2. Section 11.2 (a) Geometric Series: If | r | < 1 then X n =1 ar n - 1 = a 1 - r . (b) Contrapositive Test for Divergence: If the series X n =1 a n is convergent then lim
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Unformatted text preview: n a n = 0 3. Section 11.3 (a) The Integral Test: If f is continous, postive and decreasing on [1 , ) and let a n = f ( n ). If Z 1 f ( x ) dx is convergent then X n =1 a n is convergent. 4. Section 11.4 (a) Limit Comparison Test: Suppose that a n and b n are series with positive terms. If lim n a n b n = c &gt; 0 then either both series converges or both series diverges. 5. Section 11.5 (a) The Alternating Series Test: If X n =1 (-1) n b n where b n &gt; 0 and (i) b n +1 b n for all n and (ii) lim n b n = 0 then the series is convergent. 6. Section 11.6 (a) If a series is absolutely convergent, then it is convergent....
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