3.5. Special Sequences
Definition 3.5.3: Root of n Sequence
Root of n Sequence: This sequence converges to 1.
Root of n sequence
Proof:If n > 1, then > 1. Therefore, we can find numbers an > 0 such that
= 1 + an for each n > 1 Hence, we can raise both sides to the n-th power and use the Binomial theorem:In particular, since all terms are positive, we obtainSolving this for an we obtain0 anBut that implies that an converges to zero as n approaches to infinity, which means, by the definition of an that converges to 1 as n goes to infinity. That is what we wanted to prove.