Monotone Convergence TheoremDate: 16 Aug 1995 12:01:35 -0400 From: sbever Subject: Monotone Convergence Theorem Question: The Monotone Convergence Theorem states that any nondecreasing or increasing sequence which is bounded above converges. Also, any nonincreasing or decreasing sequence which is bounded below converges. Now, for any series for which all the terms are positive, we know that the sequence of partial sums is increasing. I assert that the Monotone Convergence Theorem can only help us conclude that the sequence of partial sums converges if we can also show that the sequence of partial sums is bounded above. Is this true? Date: 16 Aug 1995 12:08:50 -0400 From: Dr. Ken Subject: Re: Monotone Convergence Theorem Hello there! Yes, you've got it. You can make it a little bit stronger by saying that if _after_a_certain_point_ in the series all the terms are _non-negative_ then the sequence of partial sums converges if it's bounded, but the basic idea is the same. -K |
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