jgens said:With some algebra, you should be able to show that the series converges. For example, with the particular series in question we can show that:
[tex]\frac{6}{4n -1} - \frac{6}{4n + 3} = 6 \left(\frac{(4n + 3) - (4n - 1)}{(4n + 3)(4n - 1)}\right) = 6\left(\frac{4}{(4n + 3)(4n - 1)}\right)[/tex]
Using a little bit more algebra, you should easily be able to determine that the series is convergent.
Edit: Oops, somebody got here first. Sorry Dick!
If a series has a limit as n approaches infinity, it is convergent. This means that as the terms of the series continue to be added, their sum will approach a finite number.
A convergent series has a finite sum as n approaches infinity, whereas a divergent series does not have a finite sum and either approaches infinity or oscillates between values.
One example of a convergent series is the geometric series 1 + 1/2 + 1/4 + 1/8 + ... = 2, as the terms get smaller and approach 0, the sum of the series approaches 2.
The comparison test states that if a series is less than or equal to a convergent series, then the original series is also convergent. To use this test, you must find a convergent series that is larger than the original series.
Other methods include the ratio test, the root test, and the integral test. These tests involve comparing the given series to a known series or using calculus to determine the convergence of the series.