Quick series convergence question

[DE7]

Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of \sum \left(\sqrt{n+1}-\sqrt{n}\right)? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I could compare it to for the comparison test? Anyways, thanks, just need a little refresher.

 

[shmoe]

Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of \sum \left(\sqrt{n+1}-\sqrt{n}\right)? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I could compare it to for the comparison test? Anyways, thanks, just need a little refresher.

You could try the integral comparison test. Or you could try multiplying by the conjugate, this should make the series to compare with more transparent.

 

[DE7]

Still not sure what to do...multiplying by the conjugate gives me something that doesn't seem so helpful to me, as I still can't find a comparison series. The limit comparison tests don't seem to be working either. Any more hints please?

 

[DE7]

Never mind...got it.

 

[copperboy]

Add them together you'll get f(n+1)-1, it's divergence.