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A geometric series is a series with a common ratio between successive terms. It takes the form \( \frac{1}{1-r} = 1 + r + r^2 + r^3 + \ldots \), and converges when the absolute value of \( r \) is less than 1, i.e., \( |r| < 1 \). This series is quite useful for representing functions as a sum of infinite terms. In the given exercise, we use the geometric series to find the power series representation of a function by identifying a similar form in the function. The goal is to rewrite the function, \( f(x) = \frac{2}{3-x} \), to match the geometric series form \( \frac{1}{1-r} \).
By manipulating the function and factoring out constants, we transform \( f(x) \) into a new version resembling a geometric series.

The interval of convergence is crucial when working with power series, as it determines the values for which the series converges to a function. For a geometric series \( \frac{1}{1-r} \), convergence occurs for \( |r| < 1 \). Using this property helps us define the interval for the rewritten series.
In our case, with the function \( f(x) = \frac{2}{3-x} \), we rewrite it to \( \frac{2}{3(1-\frac{x}{3})} \). The geometric series for \( \frac{1}{1-\frac{x}{3}} \) converges when \( \left| \frac{x}{3} \right| < 1 \). Solving this inequality, we derive the interval of convergence as \( |x| < 3 \). Thus, the interval of convergence for the series representation is \( (-3, 3) \).

The general term of a series is a mathematical expression that represents any term in the series. It's a crucial tool because it allows us to express the entire series succinctly using summation notation.
For the function \( f(x) = \frac{2}{3-x} \), once rewritten in terms of the geometric series, the power series representation is identified as \( f(x) = \frac{2}{3} \left( 1 + \frac{x}{3} + \left(\frac{x}{3}\right)^2 + \ldots \right) \). By distributing \( \frac{2}{3} \) across the series, we express it as a sum of terms.
The general term is \( \frac{2}{3^{n+1}} x^n \). We can then write the series as: \[ f(x) = \sum_{n=0}^{\infty} \frac{2}{3^{n+1}} x^n \]. This representation is powerful for identifying patterns in each term of the series.

Series expansion refers to expressing a function as an infinite sum of terms, often involving powers of a variable. This technique is particularly useful for approximating functions and calculating values where direct computation can be complex. In simpler terms, series expansions break down a complicated function into manageable pieces.
In the exercise, we take the function \( f(x) = \frac{2}{3-x} \) and convert it into a series expansion using geometric series principles. After rewriting the function into a form suitable for the geometric series \( \frac{1}{1-r} \), we expand \( \frac{2}{3} \) times the series: \( f(x) = \frac{2}{3} + \frac{2}{3^2}x + \frac{2}{3^3}x^2 + \ldots \).

• Each term of this series is derived based on the powers of \( x \) and the general term pattern \( \frac{2}{3^{n+1}} x^n \).
• The whole expansion provides us with a deeper insight into the function, enabling us to understand its behavior over the interval \( (-3, 3) \).

This method is an excellent way to explore the characteristics of functions in calculus.