Guide for Evaluating Series
This guide will take you through the process of determining whether a series is convergent or not. When looking at a series that you want to solve, using the divergence test is pretty much the minimum requirement you have to pass in order to determine convergence. For any series, if the terms within the series do not tend towards zero, there is no possible way for the series to converge. We can check this by employing the use of a limit as n goes to infinity.
Evaluate this limit like you would any normal limit, and look at the result. With the results of your limit, it's very easy to check if a series diverges or not. If the result of the limit does not equal zero, you know that the series diverges, and you need no further test. If the result of the limit is 0 however, you do not know if the limit converges yet, but you know that it passes the first test and you can continue along with the process.
After confirming the results of the divergence test, we can employ tests of convergence on the series. The first test that we want to check is the limit comparison test. With the limit comparison test, we compare our series term with the series terms of a much more common function. The most common are the harmonic series: 1/n (diverges) and 1/n^2 (converges). We apply the limit below on one of these common functions, and check the resulting value.
There are a couple different values that c can have after taking the limit. If we result in a 0 or infinite value, you should choose a different function to compare with. If we get a positive constant however, we know that both of the series will follow the same path, either both will converge or both will diverge. This is why we choose a common function, so we can apply the convergence of the known series on the unknown. If there's no common function that you can compare with the series terms, then you can move on to the next test.
If there are no common function that we can use to represent the series, we can move along to another test, the ratio test. This test works very well with series terms that have polynomial, exponential, and factorial functions. Similar to the other test, we apply a limit on the series terms, but this time, the numerator contains our series terms with n+1 replacing n.
Just like last time, the results of this limit tell us pretty quickly if the series converges or not. If the result is less than 1, the series will always converge. If the result is greater than 1, the series will always diverge. Otherwise, the result will be exactly 1, which does not help us and we cannot conclude. In that case, we continue on with the process and try another test.
The final test covered in this guide is the integral test. The integral test involves an integral instead of a limit, but still helps us with trying to determine convergence. The integral test can tell use whether the series is convergent or not based on the results of the improper integral. In the diagram below, f(x) stands for the function that represents the series terms.
Although this function requires harder math techniques than the rest, the results are easy to look at. It is very simple once you take the integral, the series will behave exactly like the integral does. If you get a finite value for the integral, you know that the series must be convergent. On the other hand, if you get an infinite value, you know that the series must diverge.
This brings us to the end of the guide. If you are still unable to evaluate the series at this point, you might need to use a more niche test that was not covered, which you can probably find online or in your textbook.