Tests for convergence

[Emmi]

I never took HL math, but I know that one of the options covered in HL math is on series, which is what I am currently learning in my math course, so I figured I would ask you lovely HL math students. We are at the moment covering convergence tests, and I am having some difficulty in applying the tests we've learned so far to determine if a sequence converges or diverges.

The tests we have covered so far are direct comparison, limit comparison, integral test, ratio test, root test, and alternating series test.

I know with the direct comparison test one compares the sequence in question with another one and uses the other one to determine whether it converges or not based on if the larger one converges/diverges, but I often struggle with identifying a series that I can use to compare my sequence with. Is there a way to make this a bit easier?

The limit comparison test confuses me. I don't have a problem using it, but why is it that if I use this test and the limit approaches a finite number and not infinity, the series converges? Why wouldn't it be zero?

The integral test I do not have problems with.

The ratio test, root test, and alternating series test were taught quite quickly and I did not really understand them when initially covered. If someone could provide a proof and example of one or more of these tests, I would really appreciate it.

Thank you in advance, Emy Glau-ski.

[Lero]

With the direct comparison test, you are looking for a series that is always larger than the series in question. If this series that is successively larger converges, then the series that is smaller also converges, and the opposite applies with testing for divergence, but that is less common in my experience.

Often questions requiring this test need a comparison with a p-series. It's worth remembering that increasing the denominator of a fraction decreases the value of the actual term obviously. Perhaps you are asked to determine if this series converges: 1 / (n^(2) + n).

When n is positive, this series is always smaller than: 1 / n^(2). Since this series converges, as it is a p-series with p > 1, then the initial series also converges. Always focus on the highest power in the series when applying this test, and the addition of something else in the denominator makes the series smaller, which is convenient for this test.

I can't really explain the proof for the limit convergence test, nor the alternating series test. Didn't study the root test.

However, when applying the alternating series test, you are looking for two things. Firstly that the absolute value of the terms of the series are decreasing, and that the limit as n approaches infinity of the absolute value of the series is 0. This establishes conditional convergence, if the absolute value of this series converges, than the series is absolutely convergent.

A suitable example is the alternating harmonic series: (-1)^(n + 1) / n .

Firstly, the absolute value of the series is decreasing, and to prove this using the inequality: |A(n + 1)| < |A(n)|.

Secondly, the limit as n approaches infinity for the series is 0. This establishes the convergence.

The proof for the ratio test is logical, you are taking the ratio of two successive terms of a geometric series. We know from SL math that in order for an infinite series to exist, it must be that the absolute value of r < 1. That is what the ratio test is assessing, that the terms are consecutively decreasing and eventually there will be a finite sum of the infinite series.

I don't fancy explaining math with words but this will have to do

Edited November 24, 2012 by Hus

[macrofire]

1. Think of the limit comparison test as such:

If you have a series that you don't know if it converges or diverges, then you can compare it to another known series to test if it diverges or converges. By putting them in a fraction form, you can simplify the expression and see whether it converges to a finite number. If it goes to zero or infinity, then there may be a huge number in the numerator or denominator....but you don't know which series caused this to happen because to melded the two together. If it does converge to some other value, then you know that the rate of growth or decay is more or less the same. Thus you can conclude that either both converge or both diverge.

2. There is a second part to the Ratio Test.

It tells you if the series converges absolutely if L < 1 because of Hus' logic above. However, is this always the case? I don't always approve of Wikipedia, but you may use your own judgement to look at particular examples :http://en.wikipedia.org/wiki/Ratio_test

If you really need help understanding, there's a deluge of Youtube vids that provides examples. From there, you can try to recreate the logic behind the tests and even rediscover the theorems.

Roots Test:

Ratio Test: