91Ó°ÊÓ

The geometric series is a fundamental concept in calculus and sequences. It can be defined as the sum of a sequence where each term after the first is the preceding term multiplied by a constant, called the common ratio. For example, a simple geometric series can look like this:

• 1, x, x^2, x^3, ...

In mathematical terms, it is expressed as:\[\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \ldots\]where \( a \) is the first term, and \( r \) is the common ratio. For the series to converge, the absolute value of the common ratio \( |r| \) must be less than 1.
Understanding the geometric series concept is crucial when dealing with power series representations of functions. In this particular problem, the function \( \frac{1}{1-x} \) can be represented as a geometric series \( \sum_{n=0}^{\infty} x^n \), as long as \( |x| < 1 \).
This is a powerful way to express functions, giving us tools to easily handle complex fractions by breaking them into infinite series.

The interval of convergence is a range of values for which a power series converges. Understanding this interval is important because it defines where the expression for our power series accurately represents the original function.
For a geometric series, the interval of convergence is determined by the condition \( |x| < 1 \). This criterion ensures that the sum of the infinite series is finite and meaningful. For the given function \( \frac{1+x}{1-x} \), the power series remains valid within the interval below:

• \(-1 < x < 1\)

By ensuring that \( |x| \) is less than 1, we know that each term in the series becomes smaller, allowing the series to sum up to a finite value. If \( |x| \geq 1 \), the terms do not converge to zero fast enough, and the series diverges.
Understanding intervals of convergence helps in analyzing where functions can be expanded into series solutions and the limits of these expansions.

Function decomposition involves breaking a complex function into simpler parts to make calculations easier. In the context of our exercise, decomposing the function helps us derive its power series representation.
Given the function \( \frac{1+x}{1-x} \), we decompose it as follows:

• \( \frac{1}{1-x} \) — representing the standard geometric series.
• \( \frac{x}{1-x} \) — a shifted series by multiplying \( x \), which shifts the indices of the sum.

By separating these parts, each can be expressed independently as a geometric series, allowing simple computations:

• \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \)
• \( \frac{x}{1-x} = \sum_{n=1}^{\infty} x^n \)

Combining these results leads to a unified series that neatly represents the original function:\[ 1 + 2 \sum_{n=1}^{\infty} x^n \]This decomposition allows difficult problems to be solved by leveraging the properties of simpler series representations, greatly simplifying the handling of complex functions.