Calculus Definitions > Diverge Calculus

“Diverge” generally means either:

In some areas of math, diverge might simply mean “takes a different path” (for example, in KL Divergence in statistics). However, in calculus, it almost always pertains to limits or behavior of sequences and series.

## Divergent Improper Integrals

Improper integrals can be defined as a limit. Therefore, an improper integral is **divergent **if the improper integral doesn’t have a limit (i.e. the limit doesn’t exist) or if the limit tends to infinity.

## Series and Sequences that Diverge (The Divergence Test)

**Series and Sequences** can also diverge. In a general sense, diverge means that the sequence or series doesn’t settle on a particular number.

A divergent *series *will (usually) go on and on to infinity (i.e. these series don’t have limits). For example, the series

9 + 11 + 13 …

will keep on growing forever.

Not all series diverge though: some diverge **all the time**, others converge or diverge **under very specific circumstances**. For example:

- Series that diverge all the time include every infinite arithmetic series and the harmonic series.
- Series that
*sometimes*converge include the power series, which converges everywhere or at a single point (outside of which the series will diverge).

Proving divergence (or convergence) is **extremely challenging** with a few exceptions. For example, you can show that an infinite series diverges by showing that a sequence of partial sums diverges.

Series convergence tests can show you how a particular series behaves, including (click to go to that article):

- Abels’ Test
- Alternating Series Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test

## Diverge Calculus References

Boardman, M. & Nelson, R. College Calculus: A One Term Course for Students With Previous Calculus Experience. (Mathematical Association of America Textbooks).

Larson, R. & Edwards, B. Calculus of a Single Variable. 2008

Morais, J. Georgiev, and S. Sprößig, W. Real Quaternionic Calculus Handbook. 2014.