 # Diverge (Calculus)

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Calculus Definitions > Diverge Calculus

“Diverge” generally means either:

• Settles on a certain number (i.e. has a limit), or
• Doesn’t converge.

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) 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:

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. Comparison of four popular tests (Boardman & Nleson, 2015).

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

## 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.