Determine the Interval of Convergence for a Power Series: A Comprehensive Guide

Understanding the interval of convergence for a power series is a fundamental concept in calculus and mathematical analysis. This article will walk you through the process using the given example and provide a deeper understanding of the convergence criteria.

Introduction to Power Series

A power series is a series of the form:

[ sum_{n0}^{infty} a_n (x - c)^n ]

where (a_n) are the coefficients, (x) is the variable, and (c) is the center of the series. The convergence of a power series is a critical aspect of its application in various mathematical and scientific fields.

The Example: Finding the Interval of Convergence

Consider the power series defined by:

[ y 2x^{1/6} ]

Our goal is to determine the interval of convergence for this power series.

Step 1: Rewrite the Series in Terms of x

To apply the tests for convergence, it is essential to express the series in a form where (x) is the variable. In this case, we have:

[ y 2x^{1/6} ]

For simplicity, let us consider the series in the form of a power series in (x). Typically, for a series like (sum a_n x^n), we need to rewrite (2x^{1/6}) to a form that allows us to directly apply the tests.

Step 2: Apply the Ratio Test

The ratio test is a powerful tool for determining the interval of convergence. It involves taking the limit of the ratio of consecutive terms:

[ lim_{n to infty} left| frac{a_{n 1}}{a_n} right| ]

For the given series, let us assume that the series can be expressed in the form:

[ sum a_n (x - c)^n ]

Since (2x^{1/6}) is not a standard form, we will focus on the behavior of the series when (y 2x^{1/6}) is in the form (y x). Therefore, we can rewrite the series as:

[ sum a_n (2x^{1/6})^n ]

Now, applying the ratio test, we get:

[ lim_{n to infty} left| frac{a_{n 1} (2x^{1/6})^{n 1}}{a_n (2x^{1/6})^n} right| lim_{n to infty} left| frac{a_{n 1}}{a_n} cdot 2x^{1/6} right| ]

The ratio test gives the radius of convergence (R) as:

[ R frac{1}{2^{1/6}} ]

Step 3: Determine the Interval of Convergence

The interval of convergence is centered at (x 0) and extends [ R frac{1}{2^{1/6}} ] on either side. This gives us the interval:

[ -frac{1}{2^{1/6}}

Step 4: Test the Endpoints

It is necessary to check the convergence at the endpoints (x frac{1}{2^{1/6}}) and (x -frac{1}{2^{1/6}}).

For (x frac{1}{2^{1/6}}), the series becomes:

[ sum a_n left( frac{1}{2^{1/6}} right)^n sum a_n 2^{-n/6} ]

For this series, we can use the divergence test. If the terms do not approach zero, the series diverges.

For (x -frac{1}{2^{1/6}}), the series becomes:

[ sum a_n left( -frac{1}{2^{1/6}} right)^n sum a_n (-1)^n 2^{-n/6} ]

Here, the alternating series test can be applied. If the series satisfies the conditions of the alternating series test, it converges.

Conclusion

In conclusion, the interval of convergence for the given power series can be determined using the ratio test and testing the endpoints. Understanding the convergence criteria and applying these tests is essential for working with power series in mathematical analysis and other applications.

Key Takeaways

The interval of convergence for a power series is determined using the ratio test and testing the endpoints. The divergence test and alternating series test are essential for checking the convergence at the endpoints. The radius of convergence (R) is given by the limit of the ratio of consecutive terms.

References

1. Stein, R. (2016). Calculus: Early Transcendentals. W. H. Freeman.

2. Matolcsi, M. (2016). Analysis. An Introduction to Proof. World Scientific.