We are working on finding the intervals of convergence of power series in class. Why do we not have to test for the convergence of the endpoints for geometric series? They always seem to diverge. Is this a set rule when working with the geometric series for this type of problem?

When it comes to power series, the geometric series is a special case that has certain properties that make it easier to determine the intervals of convergence. One important property of the geometric series is that it converges if the absolute value of the common ratio (r) is less than 1, and it diverges if the absolute value of r is greater than or equal to 1.

To determine the intervals of convergence for a power series, which is a series of terms involving powers of a variable, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. If the limit is greater than 1 or does not exist, the series diverges.

Now, for the geometric series, the ratio of consecutive terms is always the same, which is the common ratio (r). So, when we apply the ratio test to a geometric series, we have:

|a_{n+1}/a_n| = |r|

As mentioned earlier, the geometric series converges if |r| < 1, and diverges if |r| >= 1.

Now, when it comes to testing the endpoints of the interval of convergence for a power series, we need to consider the behavior of the series at the endpoints. In the case of the geometric series, the endpoints are obtained by taking the limit as x approaches the center of the interval of convergence (c) plus or minus the radius of convergence (r).

For the geometric series, the radius of convergence is always 1. So, if we consider the endpoints obtained by plugging in x = c + 1 and x = c - 1, we get:

|r^k| = |r^1| = |r| >= 1

Since the absolute value of the common ratio (r) is greater than or equal to 1, the geometric series will always diverge at the endpoints, regardless of the value of c. Therefore, we don't have to test for the convergence of the endpoints for the geometric series.

To summarize, when working with the geometric series as a special case of power series, we do not need to test for the convergence of the endpoints because the series always diverges at the endpoints. This is due to the fact that the geometric series converges when |r| < 1 and diverges when |r| >= 1.