$\sum_{n=N}^{\infty}$

The following series will be analyzed:

$\sum_{n=N}^{\infty} \frac{\left ( x \right )^{2 n - 1}}{9^{n}}$

The following is the interval of convergence:

$\text{Click "Calculate" to begin.}$

The interval of convergence is the set of all values of $x$ for which a power series converges. It is the range of values for which the series converges to a finite value.

To find the interval of convergence, you can use the Ratio Test. Take the limit of the absolute value of the ratio of consecutive terms in the series and determine the values of $x$ for which the limit is less than 1.

Since the Ratio Test is always inconclusive at the endpoints of the interval of convergence, the convergence at the endpoints is uncertain. You need to use other convergence tests to determine the convergence of the series at the endpoints.

An inconclusive result means that the Ratio Test is inconclusive for all values of $x$. In this case, you may need to manually use other convergence tests to determine the interval of convergence. This is common in p-series and other special cases.