Find a series expansion for the expression. (x)/(1-x) for...

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Key Concepts

Geometric Series

The geometric series is a fundamental series expansion given by the formula 1/(1-x) = ? x^n for n starting from 0, provided |x| < 1. This series is central to many areas of mathematics and is used to express rational functions as infinite series, making them more accessible for analysis and computation.

Domain of Convergence

The condition |x| < 1 is critical because it ensures the convergence of the power series representation. The radius of convergence dictates the interval in which the series expansion is valid, and for the geometric series, this interval is determined by the absolute value of the common ratio being less than one.

Series Manipulation Techniques

In many series expansions, it is common to manipulate known series by operations such as multiplication by a variable. In this case, multiplying the geometric series 1/(1-x) by x shifts the series index and produces the expansion x/(1-x) = ? x^(n+1), or equivalently, ? x^n starting at n=1. This technique is a foundational tool in mathematical analysis and power series manipulation.

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