Finding the Interval of Convergence and Radius of Convergence of the Series ∑(x^n / (9n-1))

As a high school mathematics teacher, I'm often asked to help students understand the concepts of interval of convergence and radius of convergence for power series. In this article, we'll dive deep into the analysis of the series ∑(x^n / (9n-1)) and explore these important mathematical properties.

Understanding Power Series

A power series is an infinite series of the form ∑(an * x^n), where an are the coefficients and x is the variable. The series ∑(x^n / (9n-1)) is a power series with a_n = 1 / (9n-1).

Finding the Interval of Convergence

To determine the interval of convergence, we need to use the ratio test. The ratio test states that if the limit of the ratio of consecutive terms, lim(|an+1 / an|), exists and is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.

Let's apply the ratio test to our series:

lim(|an+1 / an|) = lim(|(1 / (9n+8)) / (1 / (9n-1))|) = lim(|9n-1 / 9n+8|) = 9/9 = 1

Since the limit is equal to 1, the ratio test is inconclusive. We need to use the Cauchy-Hadamard theorem to determine the interval of convergence.

The Cauchy-Hadamard theorem states that the radius of convergence, R, is given by: R = 1 / lim(|a_n|^(1/n))

Applying this to our series, we get: R = 1 / lim(|1 / (9n-1)|^(1/n)) = 1 / lim(1 / (9-1/n)) = 1 / 8

Therefore, the radius of convergence is 1/8.

The interval of convergence is (-1/8, 1/8), which includes the endpoints.

Conclusion

In this article, we have explored the concept of interval of convergence and radius of convergence for the power series ∑(x^n / (9n-1)). We used the ratio test and the Cauchy-Hadamard theorem to determine that the radius of convergence is 1/8 and the interval of convergence is (-1/8, 1/8). Understanding these properties is crucial in the study of power series and their applications in mathematics.