Why is 1n convergent

Theorem 2. I will not present the proof, but you should read the proof in the text though. It uses the binomial theorem, and it is a nice proof. This theorem is much easier to prove once we know more about increasing and decreasing bounded sequences which are covered in Section 2. Here is a Remark that is not in your textbook, but these inequalities can help you simplify equations and find nicer bounds.

This is essentially enough of an argument, but here it is more clearly written out:. Note that the above theorem does not work if the limit is not zero! Math — Convergence of Sequences of Real Numbers. Sequences Definition 2. 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. Then comes the tolerance of error which is allowed to depend on the sequence, third comes the N, which depends on the error. So now to the proof. Hopefully this is right! You are commenting using your WordPress.

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