Serving To Mathematics

There are several tests that can be used to determine the convergence or divergence of an infinite series. Some of the most basic and commonly used tests include:

The Divergence Test: If the limit of the absolute value of the nth term is not zero as n approaches infinity, then the series diverges.

The Integral Test: If the series is a non-negative term series and the function whose nth term is the nth term of the series is continuous and non-decreasing on the interval [1, infinity), then the series converges if and only if the definite integral from 1 to infinity of the function is finite.

The Comparison Test: If the absolute value of the nth term of the series is less than or equal to the absolute value of the nth term of a series that is known to converge or diverge, then the original series will converge or diverge, respectively.

The Limit Comparison Test: If the limit of the ratio of the nth term of the series to the nth term of a known series is a finite non-zero value, then the two series have the same behavior of convergence or divergence.

The Ratio Test: If the limit of the ratio of the absolute value of the (n+1)th term to the absolute value of the nth term is less than 1, then the series converges. If the limit is greater than 1, then the series diverges. If the limit is equal to 1, then the test is inconclusive and another test should be used.

The Root Test: If the limit of the nth root of the absolute value of the nth term is less than 1, then the series converges. If the limit is greater than 1, then the series diverges. If the limit is equal to 1, then the test is inconclusive and another test should be used.

It is important to note that a series may pass one test, but fail another test. Therefore, it is often a good idea to use multiple tests to confirm the convergence or divergence of a series