Tests for convergence
An infinite series converges if the sum of its terms does not approach infinity. It diverges if it does. An example is a series where every term is double its previous term. This will keep adding more and more and will diverge.
There are tests to test for convergence and divergence. The first is the nth term divergence test. If the limit as n goes to infinity of a sub n is plus or minus infinity, then the series diverges.
The p-series test is another such test. For series that can be modeled by 1/n^p if p > 1, then it converges. If p=1 then it is 1/n which is called the harmonic series. Even though it grows smaller and smaller it will never converge.
The geometric series test: If every pair of terms in a series has a common ratio r? If |r| < 1, then it converges.
The alternating series test says that if every term in a series is smaller than the previous term and it alternates positive, negative, ..., then it converges.
In the ratio test take a series evaluated at n+1 and divide it by the same series evaluated at n. If the absolute value of that is less than one it converges.
The integral test: if the integral of the series evaluated from a to infinity is not infinity it converges.
Comparison test: create a sum that is easy to find if it converges or diverges. If the original series is less than the new series and the new series converges, then the original series converges. If the original series is greater than the new series and the new series diverges, then the original series diverges.
Example: 1/(n-1) > 1/n, and 1/n diverges; therefore, 1/(n-1) diverges.
Limit comparison test: choose a function similar to the original function. Take the limit of the original function over the new function as n goes to infinity. If that is 0 and the new function converges, then the original function converges. If that is infinity and the new function diverges, then the original function diverges. If it is any value not 0 or infinity, then the original function acts the same as the new function (convergent or divergent).
There are tests to test for convergence and divergence. The first is the nth term divergence test. If the limit as n goes to infinity of a sub n is plus or minus infinity, then the series diverges.
The p-series test is another such test. For series that can be modeled by 1/n^p if p > 1, then it converges. If p=1 then it is 1/n which is called the harmonic series. Even though it grows smaller and smaller it will never converge.
The geometric series test: If every pair of terms in a series has a common ratio r? If |r| < 1, then it converges.
The alternating series test says that if every term in a series is smaller than the previous term and it alternates positive, negative, ..., then it converges.
In the ratio test take a series evaluated at n+1 and divide it by the same series evaluated at n. If the absolute value of that is less than one it converges.
The integral test: if the integral of the series evaluated from a to infinity is not infinity it converges.
Comparison test: create a sum that is easy to find if it converges or diverges. If the original series is less than the new series and the new series converges, then the original series converges. If the original series is greater than the new series and the new series diverges, then the original series diverges.
Example: 1/(n-1) > 1/n, and 1/n diverges; therefore, 1/(n-1) diverges.
Limit comparison test: choose a function similar to the original function. Take the limit of the original function over the new function as n goes to infinity. If that is 0 and the new function converges, then the original function converges. If that is infinity and the new function diverges, then the original function diverges. If it is any value not 0 or infinity, then the original function acts the same as the new function (convergent or divergent).
Radius of convergence
Often, the series will have have an x term in it such as: 2x^n. Depending on the value of x it will converge or diverge. We already know 2*1^n will diverge and 2*(-1)^n will converge. Therefore, it will converge on the interval [-1, 1). The radius of convergence is half the interval. In this case since 1-(-1) =2, r=1.
Example 1/n^(x/4) converges when p >1, so x/4 > 1, or x>4. It converges on the interval (4,infinity). This has a radius of convergence of infinity since (infinity-4)/2 = infinity.
In the first example 2x^n it converged at x=-1, but diverged at x=1. This is conditional convegrence. When a series converges at both the value of x and the absolute value of x it converges absolutely. If, like in the first example, does not, then it is said to converge conditionally.
Example 1/n^(x/4) converges when p >1, so x/4 > 1, or x>4. It converges on the interval (4,infinity). This has a radius of convergence of infinity since (infinity-4)/2 = infinity.
In the first example 2x^n it converged at x=-1, but diverged at x=1. This is conditional convegrence. When a series converges at both the value of x and the absolute value of x it converges absolutely. If, like in the first example, does not, then it is said to converge conditionally.