Definition:Divergence
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Divergent Sequence
A sequence which is not convergent is divergent.
Explain how this specifically applies in the different contexts: Topological space, metric space, etc.
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Also see
- Because a Convergent Sequence is Bounded, it follows directly that an unbounded sequence is divergent.
- However, Divergent Sequences may be Bounded shows that the converse does not necessarily hold.
Divergent Series
A series which is not convergent is divergent.
Divergent Function
A function which is not convergent is divergent.
Divergent Improper Integral
An improper integral of a real function $f$ is said to diverge if any of the following hold:
- $f$ is continuous on $[a..+\infty)$ and the limit $\displaystyle \lim_{b \to +\infty} \int_a^b f \left({x}\right) \ \mathrm dx$ does not exist,
- $f$ is continuous on $(-\infty..b]$ and the limit $\displaystyle \lim_{a \to -\infty} \int_a^b f \left({x}\right) \ \mathrm dx$ does not exist,
- $f$ is continuous on $[a..b)$, has an infinite discontinuity at $b$, and the limit $\displaystyle \lim_{c \to b^-} \int_a^c f \left({x}\right) \ \mathrm dx$ does not exist,
- $f$ is continuous on $(a..b]$, has an infinite discontinuity at $a$, and the limit $\displaystyle \lim_{c \to a^+} \int_c^b f \left({x}\right) \ \mathrm dx$ does not exist.
