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

    \displaystyle\sum_{n=1}^n \frac{1}{e^{\sqrt n}}

    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?
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    Did you try an integral test?

    e^(-r^0.5)
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    (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
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    Could use the fact that e^x goes to infinity faster than x^n for any positive power of n?
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    what about comparing with 1/e^-r? its a geometric series with common ratio 1/e, therefore it converges.
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    (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
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    (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?
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    Let a_n = e^\sqrt{n}.

    You might want to think about comparing a_{n^2} with something (and then justify what that means for a_k when k isn't a square).
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    (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.
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    Incidentally, I'm pretty sure the Cauchy condensation test works well here if you're allowed to use it.
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    (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?
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    (Original post by valhalla92)
    e^n diverges.
    yeah, that's just me being stupid
 
 
 
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