Convergence of 1/(n*n^(1/n))

  • Thread starter Piamedes
  • Start date
  • #1
41
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Homework Statement


Test for convergence the series:
[tex] a_[n] = \frac{1}{n*n^{\frac{1}{n}}} [/tex]


Homework Equations


Various Sequence Convergence Tests


The Attempt at a Solution


So far I've tried both a normal comparison and limit comparison test with n^2. The normal one seemed fine until the end. Here was my logic:

For n greater than 1 (its just less than, not equal)

[tex] n^{\frac{1}{n}} \le n [/tex]

[tex] \frac{1}{n^{\frac{1}{n}}} \ge \frac{1}{n} [/tex]

[tex] \frac{1}{n*n^{\frac{1}{n}}} \ge \frac{1}{n^2}} [/tex]

But that doesn't work because it just proves that for every term, this sequence is greater than the p-series for n^2.

For the limit comparison test I don't get an actual limit, so I can't use it.

If anyone has any suggestions for which test to use, or what series to compare it to, I would be most grateful.
 

Answers and Replies

  • #2
392
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Would [tex] n^{\frac{1}{n}} \le \log n[/tex] work?
 
  • #3
41
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uh, I'm pretty sure that your inequality is backwards
 
  • #4
392
0
uh, I'm pretty sure that your inequality is backwards

I'm pretty sure I'm right for "large" n. And on further reflection, n^(1/n) < 2 is even better.
 
  • #5
41
0
thanks, the comparison with 2 works perfectly
 

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