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A geometric series is a fascinating type of series where each term is derived from the previous one by multiplying it with a constant, known as the common ratio. Imagine you have an initial amount, the first term, and with each step, you multiply by a consistent factor to get the next term.
In more formal mathematical terms, a geometric series takes the form:

• \( \, a + ar + ar^2 + ar^3 + \ldots \)

Here, \( a \) is the first term and \( r \) represents the common ratio. In the context of the function \( f(x) = \frac{1}{1-2x} \), it reveals a geometric series.When we express it in a series, the formula is written as:

• \( \sum_{n=0}^{\infty} ar^n \)

In our case, \( a = 1 \) and the common ratio \( r = 2x \). Each term builds on the last, multiplying by \( 2x \) every time to expand the series.

Understanding when a series converges is a crucial part of solving problems involving series. Convergence means that as you add more terms in a series, they get closer and closer to a certain value and do not go off to infinity. Specifically, for a geometric series to converge, it's essential that the common ratio's absolute value is less than 1.
The condition for convergence of a geometric series \( \sum ar^n \) is:

• \( |r| < 1 \)

In the problem provided, the common ratio is \( 2x \). So, we set up the inequality, \( |2x| < 1 \), to ensure convergence. Solving this inequality by dividing both sides by 2, we find \( |x| < \frac{1}{2} \). This inequality tells us the values of \( x \) for which the series converges. When \( x \) stays within this range, the partial sums approach a specific limit, meaning the series converges.

Inequalities are quite handy in calculus, especially when dealing with series and their convergence. They help us determine the range of values for which a function behaves in a certain manner. In the analysis of geometric series, solving inequalities informs us when a series will converge.
In the exercise we're discussing, the key inequality is \( |2x| < 1 \). To solve it:

• Focus on removing absolute values: \( -1 < 2x < 1 \)
• Isolate \( x \) by dividing throughout by 2: \( -\frac{1}{2} < x < \frac{1}{2} \)

This solution signifies the interval of convergence, establishing the boundaries for the values of \( x \) in which the series converges. By respecting the solution of this inequality, you can ensure that your calculations regarding the series remain valid and the series sum approaches a definite value.