Limits of Geometric Sequence
Summary
The geometric sequence $\left\{ r^{n} \right\}$ converges to $-1 \lt r \le 1$, and its value is as follows:
$$ \lim\limits_{n \to \infty} r^{n} = \begin{cases} 0 & \text{if } -1 \lt r \lt 1 \\ 1 & \text{if } r = 1 \end{cases} $$
Proof
$r = 1$
If $r = 1$,
$$ \lim\limits_{n \to \infty} 1^{n} = \lim\limits_{n \to \infty} 1 = 1 $$
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$-1 \lt r \lt 1$
If $-1 \lt r \lt 1$, then since $| r^{n} | > | r^{n+1} |$, there exists $N$ that satisfies the following for all $\epsilon > 0$.
$$ n \ge N \implies | r^{n} - 0 | \lt \epsilon $$
Therefore, by the definition of the limit of a sequence, it is $\lim\limits_{n \to \infty} r^{n} = 0$.
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