converge absolutely or conditionally, or diverges?

[rcmango]

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

maybe alternating series, or harmonic series?

3. The attempt at a solution

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

[chanvincent]

As n-> infinity, tan(1/n) -> tan(0) -> 0
Does this help?

[quasar987]

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

\sum_{n=1}^{\infty}\tan(n^{-1})<\infty

??

[rcmango]

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.

[quasar987]

What do you mean by "continues on to infinity" ?

[mjsd]

rcmango, i think you mean using the Leibniz test (for alternating series)
there are three conditions, check all to prove.

[HallsofIvy]

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.