in a p-Series does the numerator have to equal 1, or can it be other constants?

ie does sum 6/(n^3) converge?

Thank You.

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- Dec 1st 2006, 01:45 PMredierRules for p-Series Convergence
in a p-Series does the numerator have to equal 1, or can it be other constants?

ie does sum 6/(n^3) converge?

Thank You. - Dec 1st 2006, 02:11 PMgalactus
You can take the 6 outside the sum:

If p<1, then 1-p<0, so as , therefore,

the integral converges(it's value is ) and consequently the series converges.

This was one of Euler's answers. Unfortunately, it led to nowhere.

What do we know today about ?.

*Little.*

Maybe someone else can lend some insight. - Dec 1st 2006, 02:48 PMCaptainBlack
- Dec 2nd 2006, 01:14 PMredier
So when I justify my work should I use the integral test or say it's a p-Series. Because no matter what the constant is in the numerator it will always converge, (as long as the p > 1) right?

I already used the integral test to justify it, but I was wondering if that would be required. It would be a lot less writing to just say p-Series and be done with it. I guess it would also depend on the teacher too, but I was wondering you opinions on the matter, and what you would make your students do if you were a teacher. - Dec 2nd 2006, 02:24 PMCaptainBlack
It will depend on what you are allowed to assume, so if the convergence of

p-series has been established in an earlier part of your course, then usually

you can assume it (I would also assume convergence if I had proven it in an

earlier part of the same assignment - referring back to where it was established).

RonL - Dec 2nd 2006, 02:49 PMThePerfectHacker
- Dec 2nd 2006, 03:06 PMgalactus
I know, PH, That's why I said it lead to nowhere. Maybe I should've elaborated more.