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Why 1/n converges

  1. Aug 30, 2010 #1
    Why the serie [tex]\sum\frac{1}{n}[/tex] diverges and the serie [tex]\sum\frac{1}{n^{2}}[/tex] converges? I'd appreciate an explanation beyond the definition of geometric series (I know that the sum of a geometric serie is given by a formula).

    I've found an explanation, that involves the creation of groups in the series so each of them result 1/2 (at least), so the sum diverges. Could I apply the same operation to the sum 1/n2?

  2. jcsd
  3. Aug 30, 2010 #2


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    These aren't geometric series so it will be hard to get an explanation from that direction. The integral test is a very natural way to understand why one converges and another doesn't; do you know what the integral test for convergence is?
  4. Aug 30, 2010 #3

    this converges if [tex]\alpha> 1[/tex]. In a similar way you show that the generalized harmonic series diverges for [tex]\alpha\leq 1[/tex].
  5. Aug 30, 2010 #4
    If i'm not wrong, [tex]a_{n}[/tex] converges if the integral sum of the associated function f(x), [tex]\int f(x)[/tex] has a finite valor. I think I've heard about that way of see it. Now I've realized that this is not a geometric serie. thanks
  6. Aug 30, 2010 #5
    Thanks for the explanation, I'll remember the alpha in the future.
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