Definition:Null Sequence/Real Numbers

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Definition

Let $\sequence {x_n}$ be a sequence in $\R$ which converges to a limit of $0$:

$\displaystyle \lim_{n \mathop \to \infty} x_n = 0$


Then $\sequence {x_n}$ is called a (real) null sequence.


Examples

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:

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