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The Ratio Test is a common technique used to find the convergence of series, including power series, which are infinite sums that depend on a variable, often denoted as \( x \). For a given power series \( \sum a_n(x-a)^n \), the Ratio Test involves calculating the limit of the absolute value of the ratio of successive terms:
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}(x-a)^{n+1}}{a_n(x-a)^n} \right| \]
This simplifies to:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right||x-a| \)

If this limit, L, is less than 1, the series converges absolutely. If it is greater than 1 or infinite, the series does not converge. If the limit equals 1, the test is inconclusive. This test is particularly powerful because it gives a clear condition for determining the behavior of the series based on the value of \( x \). In this exercise, the Ratio Test helps identify the values of \( x \) for which the series converges.

A series converges absolutely if the series composed of the absolute values of its terms converges. For power series like \( \sum (3^n + 2^n)(x+2)^n \), absolute convergence is established if:

• The series \( \sum |a_n(x+2)^n | \) converges.

When a power series converges absolutely, it means both that the series itself converges, and any rearrangement of its terms will also converge.
Absolute convergence is a stronger condition than regular convergence, offering more mathematical stability and robustness. In practical terms, this means we need to satisfy \( |3^n + 2^n||x+2|^n < 1 \) in the long run. Absolute convergence ensures the series behaves nicely and the Ratio Test assists in confirming it.

The convergence interval of a power series is the range of values of \( x \) for which the series converges. For the series \( \sum (3^n + 2^n)(x+2)^n \), the Ratio Test is used to determine this interval, by finding when the absolute value of the modified terms remains less than 1.
Here, we set\( 3|x+2| < 1 \) leading to \( |x+2| < \frac{1}{3} \).

• This inequality establishes the central point of convergence and the "radius," which here gives the boundary range of values.

Solving the inequality, \( |x+2| < \frac{1}{3} \) becomes \( -\frac{1}{3} < x+2 < \frac{1}{3} \) after breaking it into two parts.
Subtracting 2 from all parts, the interval transforms into \( -\frac{7}{3} < x < -\frac{5}{3} \).
This range is called the interval of convergence, and only within this interval does the series converge absolutely, confirming the results with the Ratio Test.