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# Convergence of series Watch

1. I have to decide whether the following series converges or not.

I have tried using the ratio test, but this is inconclusive. The root test does not help either. I think the comparison test would be a good bet, but i can't seem to find a suitable comparison. I'm pretty sure that the series does converge. Any suggestions?
2. Did you try an integral test?

e^(-r^0.5)
3. (Original post by valhalla92)
Did you try an integral test?

e^(-r^0.5)
we haven't done that yet, we haven't even covered integrals in analysis, so we're only supposed to use the ratio, comparison and root tests
4. Could use the fact that e^x goes to infinity faster than x^n for any positive power of n?
5. what about comparing with 1/e^-r? its a geometric series with common ratio 1/e, therefore it converges.
6. (Original post by valhalla92)
what about comparing with 1/e^-r? its a geometric series with common ratio 1/e, therefore it converges.
Presumably you mean e^-r, and unfortunately that won't work since e^(-r/2) is bigger than e^-r
7. (Original post by valhalla92)
what about comparing with 1/e^-r? its a geometric series with common ratio 1/e, therefore it converges.
That works i think, thanks wouldn't it be simpler to say compare with e^n though?
8. Let .

You might want to think about comparing with something (and then justify what that means for a_k when k isn't a square).
9. (Original post by master roflcopter)
That works i think, thanks wouldn't it be simpler to say compare with e^n though?
e^n diverges.

(Original post by IrrationalNumber)
Presumably you mean e^-r, and unfortunately that won't work since e^(-r/2) is bigger than e^-r
and yeah i meant e^-r. I kind of mixed it up through typing 1/e^r.

You got confused with the expression we're looking at. It's e^-(r^(1/2)), not e^(-r/2).

Alright, never mind, it's still larger anyway.
10. Incidentally, I'm pretty sure the Cauchy condensation test works well here if you're allowed to use it.
11. (Original post by DFranklin)
Incidentally, I'm pretty sure the Cauchy condensation test works well here if you're allowed to use it.
don't know what that is?
12. (Original post by valhalla92)
e^n diverges.
yeah, that's just me being stupid

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