So this right over here is an infinite sum or an infinite series, and what I want you to do right now is to pause this video and try to express this infinite series using sigma notation. So I'm assuming you've given a go at it, so let's just look at each term of the series and let's see if we can express it with kind of an ever-increasing index. So the first thing that might jump out at you is this oscillating sign that's happening right over here.

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Series - Basics In this section we will introduce the topic that we will be discussing for the rest of this chapter. That topic is infinite series.

So just what is an infinite series? The most common names are: You should have seen this notation, at least briefly, back when you saw the definition of a definite integral in Calculus I. If you need a quick refresher on summation notation see the review of summation notation in the Calculus I notes.

Now back to series. Had our original sequence started at 2 then our infinite series would also have started at 2. The infinite series will start at the same value that the sequence of terms as opposed to the sequence of partial sums starts. We do, however, always need to remind ourselves that we really do have a limit there!

This section is going to be devoted mostly to notational issues as well as making sure we can do some basic manipulations with infinite series so we are ready for them when we need to be able to deal with them in later sections.

First, we should note that in most of this chapter we will refer to infinite series as simply series. So for example the following series are all the same.

Do not forget however, that there is a starting point and that this will be an infinite series. Now that some of the notational issues are out of the way we need to start thinking about various ways that we can manipulate series.

We have the following properties. Each is a finite sum and so it makes the point. This is pretty much impossible since both series have an infinite set of terms in them, however the following formula can be used to determine the product of two series.

Even if both of the original series are convergent it is possible for the product to be divergent. The reality is that multiplication of series is a somewhat difficult process and in general is avoided if possible. The next topic that we need to discuss in this section is that of index shift.

The basic idea behind index shifts is to start a series at a different value for whatever the reason and yes, there are legitimate reasons for doing that. Performing an index shift is a fairly simple process to do.

There is actually an easier way to do an index shift. The method given above is the technically correct way of doing an index shift. This will always work in this manner.

Example 1 Perform the following index shifts. This final topic is really more about alternate ways to write series when the situation requires it. We could have stripped out more terms if we wanted to. A sequence is a list of numbers written in a specific order while an infinite series is a limit of a sequence of finite series and hence, if it exists will be a single value.

So, once again, a sequence is a list of numbers while a series is a single number, provided it makes sense to even compute the series. Students will often confuse the two and try to use facts pertaining to one on the other.You cannot avoid mathematical notation when reading the descriptions of machine learning methods.

Often, all it takes is one term or one fragment of notation in an equation to completely derail your understanding of the entire procedure. 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.

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In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. We will also briefly discuss how to determine if an infinite series will converge or diverge (a more in depth .

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Notation as a Tool of Thought