A test for convergence/divergence in mathematics is a method used to determine whether a mathematical series will converge (approach a finite value) or diverge (increase to infinity) as the number of terms in the series increases.
To determine if a series converges or diverges, you can use various tests such as the integral test, comparison test, ratio test, or root test. These tests involve analyzing the behavior of the terms in the series and comparing them to known patterns of convergence or divergence.
The integral test is a method for determining the convergence or divergence of an infinite series by comparing it to the integral of a related function. If the integral of the function converges, then the series also converges. If the integral diverges, then the series also diverges.
Yes, a series can have a finite number of terms and still diverge. This is known as a finite sum, and it occurs when the terms in the series do not approach zero as the number of terms increases. In this case, the series will not converge and will instead approach a finite value.
If a series does not pass any of the convergence tests, then it is considered to be divergent. This means that the series will not approach a finite value as the number of terms increases and will instead either increase to infinity or oscillate between different values.