Properties of Sequences: Survey of Methods

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There are two important properties one can ask about (other than the property - convergence): boundedness and monotonicity.

Boundedness

The easiest way to check on boundedness of a sequence is to look at the expression |a_n| and try to prove that there is a common constant h such that for all relevant n (depending where we start indexing of the sequence) one has |a_n| ≤ h.
There is no algorithm for finding such a h, all depends on experience.

Boundedness from below or from above is done similarly, one uses experience to decide whether the appropriate inequalities (a_n ≥ k for the former, a_n ≤ K for the latter) are true for some suitable k, resp. K.

Example: Explore the boundedness of the sequence .

Solution: First we attempt to prove boundedness. Is there a number h so that for all n natural one has |n² − 4n + 3| ≤ h?
The function defining our sequence is a quadratic polynomial whose graph is a parabola, in this case a parabola going up. Since we are considering natural numbers that go to infinity, from the graph it seems that the values will also increase beyond any bound. Thus the given sequence is not bounded.
However, we know that such a parabola does not extend down indefinitely. In fact, it is easy to show that its vertex has coordinates (2,−1). Thus we have the inequality n² − 4n + 3 ≥ −1 valid for all n, which means that the given sequence is bounded from below.

Again, here there is no definite algorithm for determining monotonicity. The standard procedure is as follows.

• Guess whether there is any monotonicity. Here one can use experience, it also helps to calculate the first few terms of the given sequence and make a guess based on this.
• Prove the appropriate inequality from the definition to justify your guess, most likely using some algebraic tricks.

In case the sequence is given by a function, it is possible to investigate the monotonicity of the function and with a bit of luck we get the answer and also a proof (cf. Sequences and functions in Theory - Limits).

Example: Investigate monotonicity of the sequence .

Solution: We calculate the first few terms: a₁ = 0, a₂ = 1/2, a₃ = 2/3, a₄ = 3/4. It would seem that the numbers grow, so we make a guess that the given sequence is increasing. Now we have to prove our guess:
We want to show that for all natural numbers n, the inequality a_n < a_n+1 is true. We will substitute and see:

The operations were equivalent and the last line is obviously true, hence so is the first one and the proof is finished.

Alternative solution: The sequence is given by the function f (x) = 1 − 1/x. To investigate monotonicity of this function, we find its derivative: f ′(x) = 1/x². Since this derivative is always positive, the function f is increasing on (0,∞), which includes all natural numbers; therefore also the given sequence is increasing.

Two more solved examples of sequences investigated for boundedness and monotonicity can be found in Solved Problems - Basic properties.