Sarah Ashley
Section 2B
“Use algebraic & bounded properties to deduce with proof the convergence of a sequence from the convergence of simpler sequences.”
Bounded
According to our textbook, a sequence is bounded, if there exists a real number N, such that every element of the sequence, Sn, is less than or equal to this M. This means that this number M is an upper bound on the entire sequence. There also must exist a point on the graph that none of the points below either. It helps us to think of this in such a way that –M<=Sn<M. If we want to picture this in our heads, we want to think of the sequence having both a ceiling and a floor.
Theorem 2.11 states that every convergent sequence is bounded. What this helps to us to say, is that if a sequence is not bounded, then it must not converge. It can be easy for us to see if specific sequences will be bounded or not, which will then help us to determine whether or not it will converge. An example of sequence that we could use this theorem on can be seen below.
Also part of this chapter, are 4 main theorems that are very useful when it comes to proving convergence sequences. The first of these theorems is what our book calls theorem 2.14, which is “Multiples of Limits”. This theorem deals with taking a convergent sequence Sn, and multiplying it by a constant C. This theorem allows us to conclude that if this is the case, the final answer is just the constant C multiplied by the limit of the Sn. We can also conclude that this ‘new’ sequence will also converge.
The next theorem is 2.15 which is called “Sum/Differences of Limits”. This theorem allows us to say that if Sn and Tn are convergent, then the limits of these sequences being added together will be equal to the limit of each of these separate sequences added together. This same thing holds true for the differences of limits, that if you are subtracting the limits of two convergent sequences, you can just take the limit of Sn minus the limit of Tn. Thus, this new sequence will also converge.
Theorem 2.16 is referred to as the product of limits. Again, if Sn and Tn are convergent sequences, we can simply multiply the limits of each of these sequences together. We also know that the product of these two limits will give a convergent sequence. This new sequence will also be a convergent sequence.
Similar to how we have a theorem for multiplication, addition and subtraction, we also have a theorem for division. This is theorem 2.17, which we call the Quotient of Limits. This lets us know that if Sn and Tn are both convergent sequences, the quotient will be the limit of Sn/the limit of Tn. The answer to this quotient, will be a sequence that also converges.
These theorems, once proved to be true, can be very helpful for the rest of the section on convergent sequences. Instead of having to use the definition of convergence, to check on these cases, we can simply check to see if one of these theorems will be applicable. This is a much quicker alternative to thoroughly proving convergence.
A few websites that can be used to help with this topic can be found below, along with a description of each.
http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx This website is Paul’s Online Math Notes. It goes over in much details, multiple examples of determining whether a sequence converges or diverges. Most of the ideas on this page are from Calculus, but it’s a good review of the basics.
The following website provides a proof of the theorem that if a sequence is bounded, than it must converge. https://math.stackexchange.com/questions/609030/every-bounded-monotone-sequence-converges
Another great resource for help on this topic can be found on Professor Salamone’s YouTube channel. A link to a video on this section is below. There are also some other videos on his channel that will relate to this topic as well that can be found in the stream for MATH 401. https://www.youtube.com/watch?v=CtTFYUn6ez4&list=PLL0ATV5XYF8BZx6_DvwgjgkjC2x9n4ZV-&index=23.
An example problem that we can use for this section is to prove the following: “Suppose sn is a sequence of real numbers. If sn converges to L, then the sequence tn=2sn converges to 2L.
In order to complete this problem, we must use theorem 2.14. The following work is this problem done out.
Let E>0 be arbitrarily chosen. Since sn converges to L, by the definition of convergence, there exists a natural number N in the natural numbers, such that for all n>=N: the absolute value of Sn-L is < E/2. Let n>=N be arbitrarily chose. Then, the absolute value tn-T=2Sn-2L is equal to 2(Sn-L) which is < then 2(E/2). The two’s cancel, and we are left with E which completes the proof.