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Does it converges? Or it depends on $a_{n}$ terms? I would thank to you if you can explain me if there are some cases in which the serie diverges and other in which the serie converges or if it just converges or if it just diverges.

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up vote 5 down vote accepted

It depends on $a_n$. For example, when $a_n=\frac{1}{n^2}$, it converges; when $a_n=\frac{1}{n(\log n)^2}$ for $n\ge 2$, it diverges.

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You might want to shift the argument of the logarithm in the second example a bit. As it stands $\sum_{n=1}^\infty a_n$ does not converge. –  Fabian Jan 22 '13 at 20:34
    
Thank you. I correctly thougth that it depended on $a_{n}$ but I hadn't found any example which diverge. Thank you again –  Damaru Jan 22 '13 at 21:46
    
@Fabian: In the second example, $\sum_{n=1}^\infty a_n$ converges. –  23rd Jan 23 '13 at 1:45
    
@Damaru: You are welcome! –  23rd Jan 23 '13 at 1:46
    
@richard: last time I checked $\log 1=0$ such that $a_1$ is undefined. –  Fabian Jan 23 '13 at 6:46
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