What makes a sequence diverges

Likewise, if the sequence of partial sums is a divergent sequence i. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. So, to determine if the series is convergent we will first need to see if the sequence of partial sums,. The limit of the sequence terms is,. 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. We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series. This is actually one of the few series in which we are able to determine a formula for the general term in the sequence of partial fractions. Therefore, the series also diverges.

Again, do not worry about knowing this formula. The sequence of partial sums is convergent and so the series will also be convergent. The value of the series is,. As we already noted, do not get excited about determining the general formula for the sequence of partial sums.

Two of the series converged and two diverged. Notice that for the two series that converged the series term itself was zero in the limit. 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. This one is similar to the first one. Note that the terms in this sequence alternate in signs. Sequences of this kind are sometimes called alternating sequences.

However, it does tell us what each term should be. In the first two parts of the previous example note that we were really treating the formulas as functions that can only have integers plugged into them. This is an important idea in the study of sequences and series. Before delving further into this idea however we need to get a couple more ideas out of the way. The first few points on the graph are,.

We assumed instead that you are already familiar with the concept of divergence, probably from taking calculus in the past. However we are now in the process of building precise, formal definitions for the concepts we will be using so we define the divergence of a sequence as follows. It may seem unnecessarily pedantic of us to insist on formally stating such an obvious definition.

Why do we have to go to the trouble of formally defining both of them? One way to answer that criticism is to state that in mathematics we always work from precisely stated definitions and tightly reasoned logical arguments. But this is just more pedantry. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Convergent and divergent sequences. Partial sums: formula for nth term from partial sum.

Partial sums: term value from partial sum. Practice: Partial sums intro. Infinite series as limit of partial sums. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Let's say I've got a sequence.

And it just keeps going on and on and on like this. And we could graph it. Let me draw our vertical axis. So I'll graph this as our y-axis.