In this section, we will continue to talk about properties of convergent sequences, and how to determine whether a sequence is convergent or not. We will discuss the rest of the definitions and theorems that you will need to know to determine if a sequence is convergent.
Definitions 2.24-2.27: A sequence {Sn} is monotonic if it meets any of the following properties:
{Sn} is… If…
Increasing: S1<S2<S3<…
Decreasing: S1>S2>S3>…
Non-Increasing: S1=>S2=>S3=>…
Non-Decreasing: S1<=S2<=S3<=…
This definition means that the sequence {Sn} must either be increasing or decreasing for it to be monotonic. If the sequence flip-flops between increasing and decreasing, then it isn’t monotonic. This is important because it helps prove one property of a convergent sequence. We us monotonic sequences to prove whether a sequence is bounded or not which will then tell us whether the sequence converges. These properties will allow us to use the monotone-convergence theorem to say that a sequence converges.
Theorem 2.40 (Cauchy Completeness Theorem): A sequence {Sn} of real numbers is convergent if, and only if, it is a Cauchy sequence.
This theorem means that a sequence, that contains all real numbers, will converge if a sequence is Cauchy AND that a Cauchy sequence will converge. This is important because it allows us to prove convergence by proving whether or not a sequence is Cauchy and vice versa. We can prove the forwards direction of the proof by showing the sequence converges when its a Cauchy sequence and we can prove the backwards direction by showing that a Cauchy sequence converges.
Below is an exercise problem that requires you to prove that a sequence is Cauchy by using the definition. Underneath the exercise is the solution. Try the exercise out first before looking at the solution.
http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx
This site gives a basic overview of convergence and how it relates to limits. It also gives many examples with solutions and how those solutions came about.
http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L7.html
This site splits the different properties of convergence into different sections like cauchy, monotone, convergence to reals, ect. It can help illustrate the differences between different sequences
Exercises
- If {sn} is bounded prove that {sn/n} is convergent.
Thomas, Bruckner and Bruckner; section 2.6, problem 2.6.2
2. If {sn} is a sequence all of whose values lie inside an interval [a, b] prove that {sn/n} is convergent.
Thomas, Bruckner and Bruckner; section 2.8, problem 2.8.2
3. Suppose that sn ≤ tn for all n and that sn → ∞. What can you conclude?
Thomas, Bruckner and Bruckner; section 2.8, problem 2.8.4
4. Which statements are true?
(a) If {sn} and {tn} are both divergent then so is {sn + tn}.
(b) If {sn} and {tn} are both divergent then so is {sntn}.
(c) If {sn} and {sn + tn} are both convergent then so is {tn}.
(d) If {sn} and {sntn} are both convergent then so is {tn}.
(e) If {sn} is convergent so too is {1/sn}.
(f) If {sn} is convergent so too is {(sn)2}.
(g) If {(sn)2} is convergent so too is {sn}.
Thomas, Bruckner and Bruckner; section 2.7, problem 2.7.5