Tuesday, October 6th, 2009
How to Test for Convergence Using the Alternating Series Test
Description
A detailed tutorial on testing for convergence using the alternating series test. Step by step tutorial including several examples of testing for convergence using the alternating series test for reference.
Overview
The alternating series test, like all convergence and divergence tests, is fairly easy. The hardest part is figuring out if you should use the AST, or a different test. An easy way to tell is, is the equation negative? What would happen if you pulled a negative one out? Or maybe, there is already a negative one outside of the equation. If you see any fraction, function, or any equation at all with a -1 to an odd power at the front (or at the front of the numerator, in a fraction) then you should use the alternating series test for it. If the series is decreasing over time, and the limit is approaching zero, then the series is convergent. The alternating series test is normally used in conjunction with another test for convergence.
Tags: -1, alternating, AST, Calculus, converge, convergence, decreasing, diverge, divergence, fraction, function, limit, Math, negative, one, series, test, zero
Posted in Calculus | No Comments »
Tuesday, October 6th, 2009
How to Test for Convergence Using the Geometric Series Test
Description
A detailed tutorial on how to test for convergence using the geometric series test. Step by step tutorial including several examples of testing for convergence using the geometric series test for reference.
Overview
A geometric series is a series that maintains a constant ratio between a set of terms. This series is an addition series, and would be expressed as 1/a + 1/2a + 1/4a, extending as far as you wish in either direction. If a series does not have that constant ratio, then it is not a geometric series. The series should converge at one, because as all the numbers are added they get closer and closer to one. The first term of a geometric series is given by a, and the ratio of a geometric series is given by r. If the ratio is less than one, then the geometric series converges to a / (1 – r). If the ratio is greater than or equal to one, then the series diverges. Usually the series will converge, which is why this is considered a test for convergence and not for divergence.
Tags: a, addition, Calculus, converge, convergence, diverge, divergence, equal to, first term, geometric, greater than, less than, Math, notation, r, ratio, series, summation, test
Posted in Calculus | No Comments »
