Calculus How To

Bounded Sequence: Definition

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A bounded sequence is one where the absolute value of every term is less than or equal to a particular real, positive number. You can think of it as there being a well defined boundary line such that no term in the sequence can be found on the outskirts of that line.

More formally, a sequence X is bounded if there is a real number, M greater than 0, such |xn| ≤ M for all n ε N.

The blue dots on the image below show an infinite sequence. As you can see, the sequence does not converge, for the red boundary lines never converge. However, it is bounded.
bounded sequence

Examples of Bounded Sequences

One example of a sequence that is bounded is the one defined by”

The right hand side of this equation tells us that n is indexed between 1 and infinity. This makes the sequence into a sequence of fractions, with the numerators always being one and the denominators always being numbers that are greater than one. A basic algebraic identity tells us that x-k = 1/xk. So each term in the sequence is a fractional part of one, and we can say that for every term in our sequence, |xn| ≤ 1.

Remember now our definition of a bounded sequence: a sequence X is bounded if there is a real number, M greater than 0, such |xn| ≤ M for all n ε N. Let M = 1, and then M is be a real number greater than zero such that |xn| ≤ M for all n between 1 and infinity. So our sequence is bounded.

Bounded Sequences and Convergence

Every absolutely convergent sequence is bounded, so if we know that a sequence is convergent, we know immediately that it is bounded. Note that this doesn’t tell us anything about whether a bounded sequence is convergent: it may or may not be. As an example, the sequence drawn above is not convergent though it is bounded.

Bounded Above and Below

If we say a sequence is bounded, it is bounded above and below. Some sequences, however, are only bounded from one side.

If all of the terms of a sequence are greater than or equal to a number K the sequence is bounded below, and K is called the lower bound. The greatest possible K is the infimum.

If all the terms of a sequence are less than or equal to a number K’ the sequence is said to be bounded above, and K’ is the upper bound. The least possible K is the supremum.


References

Gallup, Nathaniel. Mat25 Lecture 9 Notes: Boundedness of Sequences. Retrieved from https://www.math.ucdavis.edu/~npgallup/m17_mat25/lecture_notes/lecture_9/m17_mat25_lecture_9_notes.pdf on January 25, 2018.
Math Learning Center: Sequences. Retrieved from https://www3.ul.ie/cemtl/pdf%20files/cm2/BoundedSequence.pdf on January 26, 2018
Larson & Edwards. Calculus.


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