# Definition:Divergent (Analysis)

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## Definition

### Divergent Sequence

A sequence which is not convergent is divergent.

### 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:

$(1): \quad f$ is continuous on $\left[{a \,.\,.\, +\infty}\right)$ and the limit $\displaystyle \lim_{b \mathop \to +\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist
$(2): \quad f$ is continuous on $\left({-\infty \,.\,.\, b}\right]$ and the limit $\displaystyle \lim_{a \mathop \to -\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist
$(3): \quad f$ is continuous on $\left[{a \,.\,.\, b}\right)$, has an infinite discontinuity at $b$, and the limit $\displaystyle \lim_{c \mathop \to b^-} \int_a^c f \left({x}\right) \ \mathrm dx$ does not exist
$(4): \quad f$ is continuous on $\left({a \,.\,.\, b}\right]$, has an infinite discontinuity at $a$, and the limit $\displaystyle \lim_{c \mathop \to a^+} \int_c^b f \left({x}\right) \ \mathrm dx$ does not exist.