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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. And I'm going to graph y is equal to a sub n. And let's make this our horizontal axis where I'm going to plot our n's.
So this right over here is our n's. And this is, let's say this right over here is positive 1. This right over here is negative 1. A A sequence is a list of terms. There are main 2 types of sequence one is convergent and the other one is divergent. Convergent sequence is when through some terms you achieved a final and constant term as n approaches infinity.
Divergent sequence is that in which the terms never become constant they continue to increase or decrease and they approach to infinity or -infinity as n approaches infinity. This will always be true for convergent series and leads to the following theorem. Then the partial sums are,. Be careful to not misuse this theorem! This theorem gives us a requirement for convergence but not a guarantee of convergence.
In other words, the converse is NOT true. Consider the following two series. The first series diverges. Again, as noted above, all this theorem does is give us a requirement for a series to converge. In order for a series to converge the series terms must go to zero in the limit. If the series terms do not go to zero in the limit then there is no way the series can converge since this would violate the theorem.
Again, do NOT misuse this test. If the series terms do happen to go to zero the series may or may not converge! Again, recall the following two series,. There is just no way to guarantee this so be careful!
The divergence test is the first test of many tests that we will be looking at over the course of the next several sections. 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.
Furthermore, these series will have the following sums or values. At this point just remember that a sum of convergent series is convergent and multiplying a convergent series by a number will not change its convergence. We need to be a little careful with these facts when it comes to divergent series. Now, since the main topic of this section is the convergence of a series we should mention a stronger type of convergence.
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