Is the alternating series sum((-1)^(n-1)*(2n+1)/(n+2)) convergent?

SigurRos · Nov 21, 2005

I apologize right now for the fact that I have no idea how to use LaTeX

I can't figure out if the following alternating series is convergent or not:

Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity

the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so I have no comclusive evidence either way. Even Maple 10 couldn't give me an answer.
I have a test tomorrow. HELP!

noslen · Nov 21, 2005

Did you try the alternating series test?

uart · Nov 21, 2005

Just take a look at the limiting terms in that series. As n goes to infinity the absolute value of the terms goes to two, so alternating or not there is no way it can converge.

uart · Nov 21, 2005

BYW. I'm petty sure you can show that the above series is bounded, that is that it doesn't creep off to +/- infinity, but it definitely doesn't converge to a well defined limit point.

HallsofIvy · Nov 22, 2005

You are aware, are you not, that is a_n does not go to zero, then the series[itex]\Sigma a_n[/itex] cannot converge? That's normally the very first "series" property you learn!

Is the alternating series sum((-1)^(n-1)*(2n+1)/(n+2)) convergent?

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