Following Infinite Series, it is of interest to note the values of x that make the power series converge. This range of values is the interval of convergence. The radius of convergence is the value measuring from the center to either boundary. Usually the radius is one of three possibilities: center ± remainder, all real numbers, or center. Remainders are listed accordingly.
There are many ways to test for convergence, the simplest of which is the nth term test. The function a(n) must approach zero in order for the series to stop growing. The direct comparison test compares p(n) with another function of known result. It is almost a form of sandwich theorem. Note that p(n) must not have negative terms.
The ratio test sets two consecutive terms of p(n) as a ratio to show whether p(n) increases of decreases, which can then be used to conclude whether the series converges or not. Note again that p(n) must not have negative terms.
The absolute convergence test. Think about it.
The integral test sets the function p(n) as an improper integral. If the improper integral diverges then the series diverges (and p(n) must not have negative terms). The p series test is a strange one to conceptualize (has nothing to do with p(n)). A value of p = 1 makes the series grow incredibly slowly, and anything larger than one even by a bit makes the series decrease.
The limit comparison test is a combination of direct comparison test and ratio test.
When the series flips between addition and subtraction, it is an alternating series. The alternating series test looks for all three conditions in order for a series to be considered convergent. It is worth noting that if the alternating series does converge, the truncation error applies here as well.
There are two types of convergences. If you were to rearrange the order of the terms in an absolutely convergent series, it will not make a difference. However, the rearranging the terms in a conditionally convergent series will result in anything. You can make any sum with it~
Here is a nice systematic guide for testing convergence:
Testing convergence is like using a magical telescope to see infinity~
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