## 3.5. Special Sequences

### Definition 3.5.3: Root of n Sequence

**Root of n sequence**

### Proof:

If*n > 1*, then

*> 1*. Therefore, we can find numbers

*a*such that

_{n}> 0= 1 + afor each_{n}n > 1Hence, we can raise both sides to then-th power and use the Binomial theorem:In particular, since all terms are positive, we obtainSolving this forawe obtain_{n}But that implies that0 a_{n}aconverges to zero as_{n}napproaches to infinity, which means, by the definition ofathat converges to 1 as_{n}ngoes to infinity. That is what we wanted to prove.