So, what is this limit going to be? Example 1 Determine if the following series converge or diverge. To get the value of this series all we need to do is rewrite it and then use the previous results. So in order for a series to converge, that means that the limit, an infinite series to converge, that means that the limit, the limit, so if you’re a convergence, convergence is the same thing, is the same thing as saying that the limit as capital, the limit as capital N approaches infinity of our partial sums is equal to some finite. The following series, for example, is not a telescoping series despite the fact that we can partial fraction the series terms. When n is equal to 2, it’s 2 minus 1; it’s negative one to the first power that’s equal to that right over there. Well, this limit doesn’t exist. You give me one more, it goes from 0 to 1.
And I encourage you to pause this video and think about it, given what we see about the partial sums right over here. Example 5 Determine the value of the following series. The name in this case comes from what happens with the partial sums and is best shown in an example. So just to be clear, what this means, so the, the partial sum with just one term is just gonna be from lowercase n equals 1 to uppercase N equals 1. The partial, the partial sums of this series. Do you need additional details? What’s the limit, as N approaches infinity of S sub N.
By now you should be fairly adept at this since we spent a fair amount of time doing partial fractions seriees in the Integration Techniques chapter. Is there a finite sum that is equal to this right over here, or does this series diverge? It’s just going to be 1. S sub three, S sub three is going to be one minus one plus one.
Let me just write like this, is equal to some Finite, so Finite Value. I think yes, but no one explains this.
Divergent telescoping series
Divergent telescoping series (video) | Khan Academy
So, we can write, lets write this down so s sub n I could write it like this is going to be one if n odd it’s equal to zero if n even. The name in this case comes from what happens with the partial sums and is best shown in an example. Notes Practice Problems Assignment Problems. We will just need to decide which form is the correct form.
So, what is this limit going to be? Do you need additional details? Sign up or log in Sign up using Google. The partial, the partial sums of this series. What’s the limit, as N approaches infinity of S sub N. So, this is going to be, let’s see s sub n, if we want to write this in general terms. So capital N, so the partial sum is going to be the sum from n equals one but not infinity but to capital N of negative 1 to the n minus 1.
In order for a series to be a telescoping series we must get terms to cancel and all of these terms are positive and so none will cancel.
So just to be clear, what this means, so the, the partial sum with just one term is just gonna be from lowercase n equals 1 to uppercase N equals 1. We can now do some examples. Consider the following series written in two separate ways i. If they converge give the value of the series. Video transcript Lets say that we have the sum one minus one plus one minus one plus one and just keeps going on and on and on teoescoping that forever and we can write that with sigma notation.
If your device is not in landscape mode many of the equations will run off the side of your device should be telesdoping to scroll to see them and some of the menu items will be cut off due to the narrow screen width. So this right over here does not exist. This is now a finite value and so this series will also be convergent.
Post teelscoping a guest Name. In this section we are going to take a brief look at three special series. In that section we stated that the sum or difference of convergent series was also convergent telescpping that the presence of a multiplicative constant would not affect the convergence of a series. We can now use the value of the series from the previous example to get the value of this series.
These are nice ideas to keep in mind. Notice that every term except the first and last term canceled out.
You give me, you, you go one more, it goes from 1 to 0. The following series, for divergencr, is not a telescoping series despite the fact that we can partial fraction the series terms.
This also means that we can determine the telesco;ing of this series by taking the limit of the partial sums. We already see if s, if capital N is odd, it’s equal to 1. Actually, special may not be the correct term.