The convergence depends on the size of the exponent a:P>
- a > 0: the sequence diverges to positive infinity
- a = 0: the sequence is constant
- a < 0: the sequence converges to 0
Proof:
Write n a = e a ln(n). Then:
- if a > 0 then as n approaches infinity, the function
approaches infinity as well
- if a < 0 then as n approaches infinity, the function
approaches zero
- if a = 0 then the sequence is the constant sequence, and hence
convergent
Is this a good proof ? If not, find a better one.
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