91Ó°ÊÓ

A power series is a type of infinite series that is commonly expressed in the form \( \sum_{n=0}^{\infty} c_n (x-a)^n \). In this representation, \( x \) is the variable, \( a \) is the center of the series, and \( c_n \) are the coefficients. The given exercise involves recognizing this structure. The power series is centered at \( a = -\frac{1}{4} \), based on rewriting the series to fit \( \sum_{n=0}^{\infty} c_n (x-a)^n \).
The form of a power series allows us to analyze the values of \( x \) for which it converges. This involves determining both the radius and interval of convergence, essential characteristics of power series that tell us where the series behaves nicely, converging to a finite sum.

The ratio test is a powerful tool used to determine the convergence of a series, particularly useful for power series. For the series \( \sum a_n \), the ratio test involves evaluating the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit is less than 1, the series converges absolutely.
In the context of our power series \( \sum_{n=1}^{\infty} \frac{(4x+1)^n}{n^2} \), applying the ratio test involves the expression:

• Finding the limit
• Simplifying to \( \left| 4x + 1 \right| \)
• Determining that \( \left| 4x + 1 \right| < 1 \) for convergence

The ratio test provides a clear threshold, guiding us to determine the values of \( x \) that ensure the series converges.

Understanding the interval of convergence is critical in revealing the range of \( x \) for which the power series converges to a limit. Once you have the inequality from the ratio test, \( \left| 4x + 1 \right| < 1 \), solving it gives us the inequality for \( x \), which is \( -\frac{1}{2} < x < 0 \).
To determine full convergence behavior, analyzing the endpoints is necessary. In this exercise, you checked:

• End point \( x = -\frac{1}{2} \) yields a convergent series.
• End point \( x = 0 \) also produces a convergent series.

Thus, both ends, \( x = -\frac{1}{2} \) and \( x = 0 \), are included in the interval, giving \( \left[-\frac{1}{2}, 0\right] \) as the interval of convergence.
The radius of convergence \( R \) equates to half the length of this interval, resulting in \( \frac{1}{2} \), illustrating the intuitive understanding of convergence within this range.