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)
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.
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):
- 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.