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Converge absolutely or conditionally, or diverges?

  • Thread starter rcmango
  • Start date
  • #1
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Homework Statement



Determine if the series converges absolutely, converges conditionally, or diverges.

equation is here: http://img409.imageshack.us/img409/7353/untitledly5.jpg [Broken]

Homework Equations



maybe alternating series, or harmonic series?

The Attempt at a Solution



not real familiar with tan with series.
haven't tried much, need supporting work for the answer.
need help.
 
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Answers and Replies

  • #2
As n-> infinity, tan(1/n) -> tan(0) -> 0
Does this help?
 
  • #3
quasar987
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Actually, this says nothing at all about the series. The implication is one way only: "Sum a_n converges ==> a_n-->0" but "a_n-->0 ==> nothing".

Actually the series satisfies all the criteria corresponding to the convergence of an alternating series. Remains to see if it converges absolutely. I.e. does

[tex]\sum_{n=1}^{\infty}\tan(n^{-1})<\infty[/tex]

??
 
  • #4
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no it doesn't converge absolutely because it continues on to infinity.

however, i do ask, how do you know to test it to be less than infinity? in other words, the convergence for a alternating series passes. but what other series convergence did not pass?

so ultimately, this will converge conditionally.

for my work, i could prove this by showing the alternating series? and then showing that it also continues on to infinity?

thanks again for all the help so far.
 
  • #5
quasar987
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What do you mean by "continues on to infinity" ?
 
  • #6
mjsd
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rcmango, i think you mean using the Leibniz test (for alternating series)
there are three conditions, check all to prove.
 
  • #7
HallsofIvy
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As n-> infinity, tan(1/n) -> tan(0) -> 0
Does this help?
It is more to the point that tan(1/n) is decreasing. Knowing that it goes to 0 is neither necessary nor helpful.
 

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