## 3.5. Special Sequences

### Definition 3.5.4: n-th Root Sequence

**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

But the right-hand side approaches 1 as*1 < <**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.

*a = 0*then we are dealing with the constant sequence, and the limit is of course equal to 0.