Real Null Sequence/Examples/n^alpha x^n

Example of Real Null Sequence

Let $x \in \R$ be a real number such that $\size x < 1$.

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$\forall n \in \Z_{>0}: a_n = n^\alpha x^n$

Then $\sequence {a_n}$ is a null sequence:

$\ds \lim_{n \mathop \to \infty} n^\alpha x^n = 0$

First a lemma:

There exists $N \in \N$ such that:

$\paren {1 + \dfrac 1 N}^{\alpha + 1} \, \size x \le 1$

$\Box$

Let $x \ne 0$.

Consider the expression:

$\size {\dfrac {\paren {n + 1}^{\alpha + 1} x^{n + 1} } {n^{\alpha + 1} x^n} } = \paren {1 - \dfrac 1 n}^{\alpha + 1} \size x$

For all $n \ge N$, we have:

$\size {\paren {n + 1}^{\alpha + 1} x^{n + 1} } \le \size {n^{\alpha + 1} x^n}$

Hence for all $n \ge N$:

$\size {n^{\alpha + 1} x^n} \le \size {N^{\alpha + 1} x^n}$

So for $n \ge N$:

$\size {n^\alpha x^n} \le \dfrac 1 N \size {N^{\alpha + 1} x^N}$

$\blacksquare$