# Sequences and convergence

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Sequences are useful in a number of mathematical disciplines for studying functionsspacesand other mathematical structures using the convergence properties of sequences.

Here is a nice set of facts that govern this idea of when a rearrangement Sequences and convergence lead to a different value of a series. Nevertheless, we give here a quick definition of ergodicity for the sake of completeness.

Just like with limits of functions however, there is also a precise definition for each of these limits. First, we need to introduce the idea of a rearrangement. Also, we want to be very careful to not rely too much on intuition with these problems.

If we multiply it times negative 1 to the n, then this one would be negative and this would be positive. For example, the first four odd numbers form the sequence 1, 3, 5, 7. But then we seem like we oscillate in signs. A couple of notes are now in order about these notations.

If it converges determine its sum. So this is y is equal to a sub n. 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.

Therefore, the series also diverges. For instance, the infinite sequence of positive odd integers can be written 1, 3, 5, 7, Remember, a sub n, this is just a function of n. And I haven't rigorously defined it yet, but you can conceptualize what's going on here.

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.

And so asking what the limit of a sub n as n approaches infinity is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to?

This notation can be used for infinite sequences as well. For example, the first four odd numbers form the sequence 1, 3, 5, 7. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

Again, do NOT misuse this test. Example 1 Write down the first few terms of each of the following sequences. This notation can be used for infinite sequences as well. 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. You will need to keep track of all these tests, the conditions under which they can be used and their conclusions all in one place so you can quickly refer back to them as you need to.

The divergence test is the first test of many tests that we will be looking at over the course of the next several sections. There are times when we can i. Example 1 Determine if the following series is convergent or divergent. This theorem gives us a requirement for convergence but not a guarantee of convergence.2 Sequences: Convergence and Divergence In Sectionwe consider (inﬁnite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers.

A Convergence Test for Sequences Thm: lim n!1 ﬂ ﬂ ﬂ ﬂ an+1 an ﬂ ﬂ ﬂ ﬂ = L. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. In general finding a formula for the general term in the sequence of partial sums is a very difficult process. Convergent Sequence. A sequence is said to be convergent if it approaches some limit (D'Angelo and WestConditional Convergence, Convergent, Limit, Strong Convergence, Weak Convergence. sequences. 1, 1/2, 1/4, 1/8, Convergence of a Recursive Sequence. Mar 24,  · Calculus 2 Lecture Convergence and Divergence of Sequences.

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. Sequences and convergence
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