# Converge absolutely or conditionally, or diverges?

• rcmango

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

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

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

??

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.

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

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

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.