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April 30th, 2011, 12:11 PM  #1 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Convergence of an infinite sum
How can we determine the convergence of a sum when both the ratio and the root tests are inconclusive? Say we have the sum Wolfram Alpha gives an "approximated sum", but from the plot I can't even tell if it's convergent. 
April 30th, 2011, 12:33 PM  #2 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Convergence of an infinite sum
proglote, This particular sum is called an 'alternating series' since the terms alternate in sign. For such series, write them as where bn ? 0. If then the series converges. Sometimes we can figure out the sum by clever tricks, but most of the time we cannot get a nice closedform expression for the sum of an arbitrary series. Ormkärr 
April 30th, 2011, 01:09 PM  #3  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Convergence of an infinite sum Quote:
 
April 30th, 2011, 01:19 PM  #4 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Convergence of an infinite sum
proglote, Convergence is defined as meaning that the sum gets arbitrarily close to some definite number (its sum) as more and more terms are added. So a series like does not converge, since the partial sums are 0 and 1, depending on whether an even or odd number of terms are added, and these values certainly do not form a convergent sequence. Ormkärr 
June 20th, 2011, 12:26 AM  #5  
Member Joined: Jun 2010 Posts: 80 Thanks: 0  Re: Convergence of an infinite sum Quote:
Criteria are for example : *) converges if (?) *) if and converges, then the series in a converges too, if it's bigger than a divergent series's terms, it diverges. *) and have the same convergence.  
July 3rd, 2011, 11:29 PM  #6 
Member Joined: Jun 2010 Posts: 80 Thanks: 0  Re: Convergence of an infinite sum
For the first criterion I'm not sure about, based on the following proof, but may contain some pitfall : The series converges if since , if p=0 then it induces an hypothesis about the fact that the terms are decreasing, else , p is to determine the minimum of the terms between m and m+N, if it exists (proof?) taking the limit as , we find that tends towards 0 since N is independent of m. 
July 4th, 2011, 02:11 AM  #7 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Convergence of an infinite sum
Unfortunately your criteria don't work. A counterexample for the first is You need to converge absolutely for the second criterion. As it stands, it doesn't work since convergence does not imply absolute convergence (counterexample: For the third you need a lot more conditions on than just for integer n  otherwise we could easily have something like where for integer n but the integral clearly does not converge. Conversely, if then is finite but diverges. 
July 4th, 2011, 12:12 PM  #8 
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Convergence of an infinite sum
The series converges to around .9243 Can this be shown?. 
July 9th, 2011, 12:33 AM  #9 
Member Joined: Jun 2010 Posts: 80 Thanks: 0  Re: Convergence of an infinite sum
I don't know which series you mean. The series 1/(n*log(n)) adds up to 3.8 at 2 billions, however it's very slow. 
July 9th, 2011, 12:51 AM  #10 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Convergence of an infinite sum diverges  I don't know what all this 3.8 nonsense is about This is a case of where you can use the corresponding integral to check convergence  this is because the integrand is monotonic. 

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