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Convergance of a series - showing limit is irrational

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Hi guys

Question is above. So I've proven that series is convergant (2nd line) by using the fact that a_n < n, and that yields that the series is less than (1 + e) as n tends to infinity, hence converges (by comparison test).

Now, the next part is where I am stuck. Using the conditions they give, I can only show the series is less than a finite value (that is irrational).... but that doesn't help proving it's irrational itself.

Any help is much appreciated.
Reply 1
Well, one direction is fairly easy. If it is not true that a_n is <= n for infinitely many n, then a_n = n-1 for infinitely many n, and also a_n < n-1 for only finitely many n. You can do a similar thing to the other condition (a_m > 0).

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