Real Null Sequence/Examples

Examples of Real Null Sequences

Example: $n^\alpha x^n$

Let $\alpha \in \Q$ be a (strictly) positive rational number.

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:

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

Example: $\dfrac {n^\alpha} {y^n}$

Let $\alpha \in \R$ be a (strictly) positive real number.

Let $y \in \R$ be a real number such that $\size y > 1$.

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

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

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

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