Calculus How To

Increasing Sequence

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An increasing sequence increases as you travel along the number line.
In notation:

  • a1 ≤ a2 … or
  • an ≤ an+1 [1].

In other words, the second term is larger than or equal to the first term, the third term is larger or equal to the second, and so on.


The above definition poses a small problem in that it can define a sequence where the numbers are all equal to each other. When this happens, it’s usually called a constant sequence instead. In order to be a true “increasing sequence”, at least one of the terms in the sequence has to be larger than the one before it. For example:

  • Constant sequence: {1, 1, 1, 1, 1}
  • Increasing sequence: {1, 1, 1, 1, 2}

Some authors use the term non-decreasing sequence to describe an increasing sequence and strictly increasing to describe increasing sequences. This may be more intuitive, because a constant sequence is certainly non-decreasing. However, all the different terminology can get a little confusing; Make sure you understand exactly what definition your textbook author (or professor) is using.

Monotonically Increasing / Strictly Increasing Sequence

At first glance, the definition for a monotonically (or monotone) increasing sequence looks exactly the same as the one for the plain old “increasing” one:

A sequence of natural numbers is monotone increasing if an ≤ an + 1 [2].

Again, the confusion here is that there really isn’t a difference between the two definitions. Some authors simply use the term “monotonically increasing” to distinguish it from strictly increasing sequences. Strictly increasing sequences always increase from term to term; monotonically increasing sequences can stay constant somewhere— they don’t always have to increase. For example:

  • Strictly increasing: {1, 2, 3, 4, 5}
  • Monotone increasing: {1, 2, 2, 3, 4}

Ascending and Descending Sequences

If you can stand a little more confusion, non-decreasing sequences are also called weakly increasing sequences. Edsger Wybe Dijkstra, one of the most influential members of computing science’s founding generation, suggested that a better replacement for both terms is “ascending”. Imagine climbing (ascending) a stair case. It’s certainly possible to take a breather here and there (stay constant), and you can’t go down (in that case, you would be descending). so Professor Dijkstra’s alternative definition makes a lot more sense.

References

[1] Zhou, A. Monotone Sequences. Retrieved April 7, 2021 from: https://www.math.ucla.edu/~azhou/teaching/19S/131-week-05.pdf
[2] Goodall, A. Mathematical Analysis I. Retrieved April 7, 2021 from: https://iuuk.mff.cuni.cz/~andrew/MAex4s.pdf
[3] Dijkstra, E. Largely on nomenclature.
https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/ewd768.html

CITE THIS AS:
Stephanie Glen. "Increasing Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/increasing-sequence/
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