Real Null Sequence/Examples
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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:
- $\ds \lim_{n \mathop \to \infty} \dfrac {n^\alpha} {y^n} = 0$