December 17th, 2011, 05:51 PM  #1 
Senior Member Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0  Series Divergence Theorem
I have a question about the divergence theorem that has confused me for some time. The divergence theorem says This seems to be contradicted in the example for a telescoping series below. Here they say that goes to 1, and the series sum is one. 
December 17th, 2011, 05:55 PM  #2 
Senior Member Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0  Re: Series Divergence Theorem I don't see the contradiction. 
December 17th, 2011, 05:59 PM  #3 
Senior Member Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0  Re: Series Divergence Theorem
here 
December 17th, 2011, 06:02 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Series Divergence Theorem
But and as wnvl pointed out, . However, and so . You are confusing with . 
December 17th, 2011, 06:15 PM  #5 
Senior Member Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0  Re: Series Divergence Theorem
ok, I think I see now. is the sequence of terms. The Nth them must be zero otherwise you will never stop adding to the series, and it will have to diverge. is the partial sum of the first nterms. It is confusing because all we are doing is putting in a another form and saying now it's now zero it's one. 
December 17th, 2011, 06:27 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Series Divergence Theorem
Yes: So, by the theorem cited above, we know the infinite series is divergent if . This does not mean necessarily however, that the infinite series converges if . Once we've established it diverges then we are done, but if the test for divergence fails, then we must prove whether it converges or not through the various tests for convergence. 

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