91Ó°ÊÓ

The interval of convergence is essentially the range of values for \( x \) that will result in a converging power series. In this exercise, the power series is centered at \( x = 1 \). To find the interval, the Ratio Test helps determine where the series converges by simplifying a mathematical expression to \( \left| \frac{x-1}{3} \right| < 1 \).
This inequality can be rewritten to find the values of \( x \). By solving \(-3 < x-1 < 3\), we arrive at \(-2 < x < 4\).
So, the series converges when \( x \) is between \(-2\) and \(4\), giving us an interval of \((-2, 4)\). This interval gives a broader understanding of where the series behaves nicely, converging to a finite value.

Absolute convergence is a stronger form of convergence. A series is said to converge absolutely if the series of its absolute values converges. For our example, we check the endpoints \( x = -2 \) and \( x = 4 \) because our interval of convergence was found to be \((-2, 4)\). Repeat all endpoints that were skipped earlier from the ratio inequality.
At \( x = 4 \), the original series becomes a p-series represented by \( \sum_{n=1}^{\infty} \frac{1}{n^3} \). Since the value of \( p = 3 \) is greater than 1, we know it converges absolutely.

Absolute convergence implies that not only does the series sum to a finite value, but it does so regardless of the order of its terms. It’s useful since absolutely convergent series can often be manipulated just like finite sums.
Knowing absolute convergence is important, particularly for rearranging terms without affecting the sum.

Conditional convergence is a bit more delicate than absolute convergence. A series is conditionally convergent when the series itself converges, but the series of absolute values diverges.
For our problem, at \( x = -2 \), the series becomes \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} \).
This is an alternating series, and by the alternating series test, it converges.

• The terms decrease in absolute value, \( \frac{1}{n^3} \) as \( n \) increases.
• The limit of the terms as \( n \to \infty \) is 0.

However, the series of absolute values, \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), diverges. This means the original series converges only because of the alternation of positive and negative terms. Conditional convergence is sensitive to the order of terms, unlike absolute convergence. This makes it critical in situations where reordering terms could lead to different sums.