Why does 1/n^2 converge

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 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. Absolute convergence is stronger than convergence in the sense that a series that is absolutely convergent will also be convergent, but a series that is convergent may or may not be absolutely convergent.

When we finally have the tools in hand to discuss this topic in more detail we will revisit it. The idea is mentioned here only because we were already discussing convergence in this section and it ties into the last topic that we want to discuss in this section.

First, we need to introduce the idea of a rearrangement. A rearrangement of a series is exactly what it might sound like, it is the same series with the terms rearranged into a different order. The issue we need to discuss here is that for some series each of these arrangements of terms can have different values despite the fact that they are using exactly the same terms. The values however are definitely different despite the fact that the terms are the same.

Here is a nice set of facts that govern this idea of when a rearrangement will lead to a different value of a series. Christian Hein 3 3 3 bronze badges. Nice telescope, Galileo. Tanner 1. Jyrki Lahtonen Jyrki Lahtonen k 21 21 gold badges silver badges bronze badges.

I'd like to see others like this if there is some central repository I reproduced this with Mathematica. First I had no borders at all, then very thin ones, then way too thick.

This is a compromise. I agree with you in the sense that the next stack would look horrible with this border thickness. Too bad I already closed the notebook. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Linked Related 0. Hot Network Questions. Mathematics Stack Exchange works best with JavaScript enabled.