Convergence+Testing

__**Convergence Testing**__

Convergence testing is used to test the endpoints of the interval of convergence of any power series or to test the convergence of any infinite series.

Consider the following infinite series:



Now using the Ratio Technique we find the interval of convergence









Now this process tells us that the series converges for any value of x between 2 and 8. It doesn't tell us, however, if the series converges for the values that contain the interval (i.e. the endpoints), in this case 2 and 8. Convergence testing is used to determine whether the endpoint values make the series converge or diverge. There are several tests used to test the endpoints but not all of them are suitable for all kinds of series. Some tests are only appropriately used on certain kinds of series. In this wiki, you'll learn what the tests are, how to use them and when to use them.

Remember that when testing endpoints you must plug the appropriate number in for x to get a series in terms of n. in the example above, to find out whether the series converges or diverges at 2, you would plug 2 in to get the series, thus getting:



This would be the series you will be running the tests on. You would then repeat the process with the value 8 (or vice versa if you began with 8).

This is a list of the the tests used in convergence testing:



And here is a flow chart that shows when and in what order each of the tests should be used:

http://www.math.psu.edu/files/141series.pdf = = =__**When to use the Tests:**__=

Also remember that if for any series then then it diverges and there is no need for further testing.

Here are some examples of when to use some of the tests:

//**Alternating series:**// If you are given an equation that looks like you should know to use the alternating series test since the (-1) to the power of n will surely make it alternate. If we write out some terms we get -(3/5)+(1/3)-(3/13). The series alternates and the absolute value of each term decreases so it converges by the alternating series test.

//**Geometric series:**// Some series look geometric and can easily be tested. These are always a coefficient multiplied by a ratio to the power of n. An example would be: .. The series is geometric because it fits the previous descriptions. It converges because is is geometric and the ratio is less than 1 (in this case ).

//**Integral test:**// If you can integrate a function easily, use the integral test. For example, the series can be integrated easily to. If the integral converges, so does the series.

//**Telescoping series:**//

If you are stuck on one of these and don't know what to do, try whiting some terms out, you might get an alternating series or a telescoping series. For example, if you get a series who's terms look similar to you can tell that the  in the first set of parenthesis cancels the  in the third set of parenthesis. The in the second set cancels the one in the fourth, and so on. The only number remaining is, therefore the series converges to that number. Keep in mind that if the series telescopes but does not pass the nth therm test, it is still divergent.


 * //Harmonic series://**

Harmonic series always diverge. This can be demonstrated by comparing the limits (using the limit comparison test) of any harmonic series with he divergent p-series.


 * //Ratio test://**

Sometimes you will find series which contain factorials. In this case, it is best to use the ratio test because it cancels out the factorials. The only problem with this test is that is may come up inconclusive (if is gives you 1 then it is inconclusive and you need to use another test).

//**Limit comparison test:**//

The limit comparison test can be used to solve the convergence of messy series (like quotients with polynomials for example). If the given series is, then you can compare its limit to the limit of the series. You can do this because the variable with the highest exponents in the numerator and denominator are kept. The test itself involved finding the limits of the series 1 (a) and 2 (b) and dividing them:. In this case we get. is a positive and finite, meaning that the series being tested has the same convergence properties at the series use as a comparison. We know that diverges (it's a harmonic series); therefore  diverges too.

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Citation: Stewart, James. "Single Variable Calculus: Fourth Edition"