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Definition 3.5.4: n-th Root Sequence

n-th Root Sequence: This sequence converges to 1 for any a > 0.

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n-th Root sequence with a = 3

Proof:

Case a > 1:
If a > 1, then for n large enough we have 1 < a < n. Taking roots on both sides we obtain
1 < <
But the right-hand side approaches 1 as n goes to infinity by our statement of the root-n sequence. Then the sequence {} must also approach 1, being squeezed between 1 on both sides (Pinching theorem).
Case 0 < a < 1:
If 0 < a < 1, then (1/a) > 1. Using the first part of this proof, the reciprocal of the sequence {} must converge to one, which implies the same for the original sequence.
Incidentally, if a = 0 then we are dealing with the constant sequence, and the limit is of course equal to 0.

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