# Definition:Divergent (Analysis)

## 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 $\hointr a \to$ and the limit $\ds \lim_{b \mathop \to +\infty} \int_a^b \map f x \rd x$ does not exist
$(2): \quad f$ is continuous on $\hointl \gets b$ and the limit $\ds \lim_{a \mathop \to -\infty} \int_a^b \map f x \rd x$ does not exist
$(3): \quad f$ is continuous on $\hointr a b$, has an infinite discontinuity at $b$, and the limit $\ds \lim_{c \mathop \to b^-} \int_a^c \map f x \rd x$ does not exist
$(4): \quad f$ is continuous on $\hointl a b$, has an infinite discontinuity at $a$, and the limit $\ds \lim_{c \mathop \to a^+} \int_c^b \map f x \rd x$ does not exist.