Converge absolutely or conditionally, or diverge?

[rcmango]

Homework Statement

heres the problem: http://img145.imageshack.us/img145/161/55180818ir8.png

does this series converge absolutely, conditionally, or does it diverge?

Homework Equations

The Attempt at a Solution

whats series to test it with?
at first glance it looks like an alternating series. reminds me of a failing p series in the denominator.

 

[HallsofIvy]

The very, very simplest of convergence tests: if a series converges the individual terms must go to 0. Does that happen here? What is
$$lim_{n\rightarrow \infty} \frac{(-1)^n}{10^{1/n}}$$?

 

[rcmango]

Okay, it appears to be alternating, however it is not decreasing because of the denominator, it is increasing!

thus diverging by the alternating series test, correct?

 

[quantumdude]

What he's using there is the nth term test for divergence. It should have been the first test that you learned, no?

 

[rcmango]

yes, an easy test, how do i take the nth term test to a variable with a n root? the 10^1/n is confusing me. please help.

 

[HallsofIvy]

No, what I said was "If the individual terms do not go to 0, then the series must diverge"!

What is the limit of $$\frac{(-1)^n}{10^{(1/n)}}$$? Is it 0?

If you have trouble with that, what about $10^{1/n}$ itself? Calculate a few values, say n= 10 and higher.

 

[rcmango]

i definitely can see that this does not go to 0.

However, how do I prove this on paper?

It fails convergence obviously.

Can i use direct comparison with harmonic series, and say that the harmonic series is smaller or equal to this series, thus showing if the harmonic series diverges, then the original series diverges.

 

[quantumdude]

However, how do I prove this on paper?

Just take the limit of the summand as n goes to infinity. You don't need to compare the series with anything.