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    Hi, I have a small test tomorrow for my Analysis module, partially about ratio tests for sequences, and I'm being asked to prove this:

    "Suppose an is a sequence such that (an+1/an) tends to L

    If L > 1 and an > 0 for all n in the natural numbers, then (an) tends to infinite"

    Yet I have no idea how to do this. Can anyone help?
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    (Original post by CraigWM)
    Hi, I have a small test tomorrow for my Analysis module, partially about ratio tests for sequences, and I'm being asked to prove this:

    "Suppose an is a sequence such that (an+1/an) tends to L

    If L > 1 and an > 0 for all n in the natural numbers, then (an) tends to infinite"

    Yet I have no idea how to do this. Can anyone help?
    Note: Always best to post in the maths subforum, rather than this umberlla one. My analysis is rusty, and there are way better analysts on the maths forum.

    Here's an outline:

    Since limit L is > 1, there's going to be a point, N, beyond which our ratio is > (L+1)/2 say (I choose a point halfway between 1 and L, and lets call it "m").

    Since a_n is >0, we have a_{n+1}>=ma_n for n>=N

    (may need a line or two here)

    and it follows that a_{N+t}>=m^ta_N for all positive integer t.

    Now take the limit as t goes off to infinity, bearing in mind a_N is positive and m > 1.
 
 
 
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