In other words, a sequence is strongly mixing if and only if the two random vectors and tend to become more and more independent by increasing for any.
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A sequence is nothing more than a list of numbers written in a specific order. The list may or may not have an infinite number of terms in them although we will be dealing exclusively with infinite sequences in this class. In the notation above we need to be very careful with the subscripts.
This is an easy mistake to make when you first start dealing with this kind of thing. There is a variety of ways of denoting a sequence. Each of the following are equivalent ways of denoting a sequence. A couple of notes are now in order about these notations.
First, note the difference between the second and third notations above. If the starting point is not important or is implied in some way by the problem it is often not written down as we did in the third notation.
A sequence will start where ever it needs to start. Example 1 Write down the first few terms of each of the following sequences. Sequences of this kind are sometimes called alternating sequences. However, it does tell us what each term should be. Before delving further into this idea however we need to get a couple more ideas out of the way.
It is the same notation we used when we talked about the limit of a function. Using the ideas that we developed for limits of functions we can write down the following working definition for limits of sequences.
The working definitions of the various sequence limits are nice in that they help us to visualize what the limit actually is. Just like with limits of functions however, there is also a precise definition for each of these limits.
Mar 24, · Calculus 2 Lecture Convergence and Divergence of Sequences. A limit of a sequence of points One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: A sequence of real numbers is convergent if and only if it is a Cauchy sequence. This . Examples and notation. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are .
Now that we have the definitions of the limit of sequences out of the way we have a bit of terminology that we need to look at. So just how do we find the limits of sequences? Most limits of most sequences can be found using one of the following theorems.
We will more often just treat the limit as if it were a limit of a function and take the limit as we always did back in Calculus I when we were taking the limits of functions. So, now that we know that taking the limit of a sequence is nearly identical to taking the limit of a function we also know that all the properties from the limits of functions will also hold.
Next, just as we had a Squeeze Theorem for function limits we also have one for sequences and it is pretty much identical to the function limit version.
This will be especially true for sequences that alternate in signs. The following theorem will help with some of these sequences. In this case however the terms just alternate between 1 and -1 and so the limit does not exist.
Example 2 Determine if the following sequences converge or diverge. If the sequence converges determine its limit.
Also, we want to be very careful to not rely too much on intuition with these problems. We will need to use Theorem 2 on this problem. So, by Theorem 3 this sequence diverges. We now need to give a warning about misusing Theorem 2.
Theorem 2 only works if the limit is zero. If the limit of the absolute value of the sequence terms is not zero then the theorem will not hold. The last part of the previous example is a good example of this and in fact this warning is the whole reason that part is there.
So, be careful using this Theorem 2.Learn about some of the most fascinating patterns in mathematics, from triangle numbers to the Fibonacci sequence and Pascal’s triangle.
2 Sequences: Convergence and Divergence In Section , we consider (inﬁnite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. focus on the basic terminology and convergence of sequences More on Sequences – Here we will take a quick look about monotonic and bounded sequences. Series – The Basics – In this section we will discuss some of the basics of infinite series. Get the free "Sequence Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.
A Convergence Test for Sequences Thm: lim n!1 ﬂ ﬂ ﬂ ﬂ an+1 an ﬂ ﬂ ﬂ ﬂ = L. Convergent Sequence. A sequence is said to be convergent if it approaches some limit (D'Angelo and West , Conditional Convergence, Convergent, Limit, Strong Convergence, Weak Convergence.
sequences. 1, 1/2, 1/4, 1/8, Convergence of a Recursive Sequence. A limit of a sequence of points One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: A sequence of real numbers is convergent if and only if it is a Cauchy sequence.
This . Mar 24, · Calculus 2 Lecture Convergence and Divergence of Sequences. Get the free "Sequence Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle.
Find more Mathematics widgets in Wolfram|Alpha.