Interval+of+Convergence+Using+Ratio+Technique

__**Interval of Convergence Using the Ratio Technique**__

The interval of convergence for an infinite series is the set of x values for which a series converges to a finite value. In order to solve for this interval, one must apply the ratio test.

ratio test:

For the series,if L=, then

i) The series converges absolutely if L<1. ii) The series diverges if L>1 iii)The series may either converge or diverge if L=1

Thus, to find an interval of convergence for x, you set up an inequality, L<1. This would look like<1.

In practice this looks like:

Find the complete interval of convergence of the power series

Here is a video explaining how to solve for the open interval of convergence:

media type="file" key="Interval1.wmv" width="356" height="356"

Once you find the open interval of convergence, you'll need to test the endpoints: media type="file" key="Interval2.wmv" width="360" height="360" Here is the work done for the interval of convergence in this video:

**Interval of convergence for lnx**




The more terms you add to the series, the series becomes more accurate on the interval of convergence. So, for lnx, the series would become more accurate on the interval (0,2] when you add more terms to the series. For instance, an approximation of ln(1.5) would become more accurate with a natural log series with 20 terms, as opposed to one with 3 terms. An approximation for ln(3), however, would not be accurate at all because 3 is outside the interval of convergence.

Basically you'll want to use the ratio test to test for the open interval of convergence, then test the endpoints. Below, a link to convergence testing of series is included. Convergence Testing

http://www.wolframalpha.com/input/?i=ln(x)+series&asynchronous=false&equal=Submit http://en.wikipedia.org/wiki/Taylor_series