Monotone Convergence Theorem

Date: 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