Sequences, series, summability Definition

Sequence tending to infinity

Definition

Let $(a_n)$ be a real sequence. The sequence tends to $+\infty$ if, for each real number $M$ , there is a natural number $N$ such that $a_n > M$ for all $n \ge N$ . Then the notation is $a_n \to +\infty$ as $n \to \infty$ , or $\lim_{n \to \infty} a_n = +\infty$ . The sequence tends to $-\infty$ if $-a_n \to +\infty$ .

Remarks

The sequences $n$ , $n^2$ and $2^n$ tend to $+\infty$ . A sequence that tends to $+\infty$ is not bounded above, so it is a divergent sequence. The converse fails. The sequence $0, 1, 0, 2, 0, 3, \dots$ is not bounded above, but it does not tend to $+\infty$ .

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Sequence definition
Real sequence definition
Sequence tending to infinity definition

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Tag	`09AZCV6GMC`
URI	`m2://09AZCV6GMC`
Kind	definition
Writing	W0 stub
MSC	40A05
Defines	Sequence tending to infinity
Uses	Real sequence

Sequence tending to infinity

Definition

Remarks

Reading path

Where this leads

Further reading

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