May 1st, 2013, 02:02 PM  #1 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  series divergence
How can I prove that the series diverges?

May 1st, 2013, 10:31 PM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: series divergence [color=#000000], decreases with , using the "Leibniz test" we deduce that the series diverges.[/color] 
May 3rd, 2013, 03:16 AM  #3 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  Re: series divergence
Good afternoon Zardo, Leibniz test states that if is a nonincreasing sequence with , then the series converges. But the theorem don't say anything about the case is false. In other words, if is false, the series not necessarilly diverges. The theorem is an implication , not an equivalence "if and only if". Do you agree? 
May 4th, 2013, 11:01 AM  #4 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: series divergence [color=#000000]Yes you are right! Flip of the mind . Here is another idea, suppose that and that then on the other hand , but this is a contradiction since our hypothesis was that and this means that the series diverges. Here is the funny part of this problem, I used mathematica to compute this series and it gave the following result , but on the other hand (which is the same) gave the result "the series does not converge"! Weird huh? This means that when trying to compute series with numerical methods, one should be very careful. [/color] 
May 4th, 2013, 07:34 PM  #5 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  Re: series divergence
Thank you, George. I wish you a nice weekend.


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