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May 28th, 2015, 06:56 AM   #1
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Series convergence

Hi,
I would appreciate a little help with saying something about convergence of the following series (sorry for not using LaTeX):

a_n=((-1)^n)(e^(1/n)-1)((n^2+4)^(1/2)-(n^2+n)^(1/2)).

I have found it is not going to converge absolutely (comparing with 1/n).
Next I used criterion of Leibniz (I dont know the proper name of this criterion or rule). So lim(n->inf)(abs(a_n))=0 but I have problem with proving that for every integer a_n >= a_(n+1).

Thank you in advance for help or showing me a way how to continue.
PS: Sorry for bad english
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May 28th, 2015, 01:13 PM   #2
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I don't know if this would satisfy you. However for large n, $\displaystyle |a_n| \approx \frac{1}{2n}$.
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