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April 30th, 2011, 12:11 PM   #1
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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.
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April 30th, 2011, 12:33 PM   #2
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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 closed-form expression for the sum of an arbitrary series.

-Ormkärr-
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April 30th, 2011, 01:09 PM   #3
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Re: Convergence of an infinite sum

Quote:
Originally Posted by Ormkärr
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 closed-form expression for the sum of an arbitrary series.

-Ormkärr-
These two conditions are met, but do they ascertain that the series converge to a determinate number? Can't the sum be undefined in an interval, the same way as -1 + 1 - 1 + 1..?
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April 30th, 2011, 01:19 PM   #4
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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-
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June 20th, 2011, 12:26 AM   #5
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Re: Convergence of an infinite sum

Quote:
Originally Posted by proglote
How can we determine the convergence of a sum when both the ratio and the root tests are inconclusive?

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.
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July 3rd, 2011, 11:29 PM   #6
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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.
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July 4th, 2011, 02:11 AM   #7
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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.
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July 4th, 2011, 12:12 PM   #8
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Re: Convergence of an infinite sum

The series converges to around .9243

Can this be shown?.
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July 9th, 2011, 12:33 AM   #9
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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.
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July 9th, 2011, 12:51 AM   #10
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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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