I'm studying infinite series and am really struggling with memorizing all the tests for convergence in my book, there's like 10 of them. I don't think I'm going to be successful in memorizing all of them. I will never be asked in my course to use a specific test to determine convergence but to determine weather a series converges using a test and showing my work.
I'm really afraid that on my examinations there are going to be series were the only way I can evaluate them is by using one test only, one that I learned that is, or that I'm going to get my examination and see a question that asks me to determines weather or not a infinite series converges or not, and my mind will just go blank as I go through all the tests that I know in my head, which will be hard to do when I don't really have them all memorized...
So I was wondering if someone could inform me of the be all end all test for convergence for infinite series, that can be used for any infinite series, or most of them. I wouldn't mind spending some time learning something new, if I can just really get the one technique down and master it. There's got to be some test that can be done on all series, or most of them, that's in some upper math courses that I haven't studied yet.
The problem here is that a method that always works is usually less efficient. The tests for convergence can all be derived from first principles using the rigorous definition of a limit.
The best thing you can do is to study the derivation of all the tests and also to solve some problems without using any of the standard test from first principles. You'll then get a better feeling of when to apply which test.
The general method that is practical for summations is to use the Cauchy property. So a series converges if and only if for every epsilon > 0 there exists an N such that for all n and m larger than N, the absolute value of the summation from n to m is less than epsilon.
It's great that you are looking for a general test that can be used for convergence of infinite series. While there is no single test that can apply to all series, there is a powerful and widely used test called the Ratio Test that can help determine convergence of many series.
The Ratio Test can be applied to a series $\sum a_n$, where the terms $a_n$ are non-zero. To use the Ratio Test, you need to take the limit of the absolute value of the ratio of consecutive terms:
$$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$
The Ratio Test has three possible outcomes:
1. If the limit is less than 1, $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, then the series is absolutely convergent. This means the series converges regardless of the signs of its terms.
2. If the limit is greater than 1, $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1$, then the series is divergent. This means the series does not converge.
3. If the limit is equal to 1, $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1$, the Ratio Test is inconclusive. In such cases, the test does not provide enough information to determine the convergence or divergence of the series.
It is important to note that the Ratio Test doesn't work for every series, but it is a very useful tool for many series encountered in calculus and beyond. When faced with a series problem, applying the Ratio Test can be a good starting point to determine convergence. However, if the Ratio Test is inconclusive, you might need to explore other convergence tests specific to certain types of series.
Remember, though, it is still important to have a good understanding of other convergence tests, as different tests may be better suited for specific series. Having a variety of tools in your arsenal will help you tackle a wider range of problems. Practice and familiarity with various tests will make it easier for you to determine when to apply each test.
I hope this explanation helps you in your studies. Good luck with your infinite series!