91Ó°ÊÓ

In the context of a geometric series, **convergence** refers to whether the series approaches a specific value as more terms are added. For a geometric series, convergence is determined by the **common ratio** \( r \). The series will converge if the absolute value of this ratio is less than one.
This is expressed mathematically as:

• \( |r| < 1 \)

If this condition holds, the series will sum to a finite value. Otherwise, the series is considered divergent, meaning it does not settle toward a specific number as more terms are added.
In the given exercise, our series has a ratio of \( r = \frac{y}{3} \). Therefore, the series converges if \(|y| < 3\). This criterion helps determine the acceptable range of values for \( y \), over which the series will result in a reliable sum and not stretch indefinitely.

Once we establish that a geometric series converges, we can find its sum. The formula for the sum \( S \) of an infinite converging geometric series is given by:

• \( S = \frac{a}{1 - r} \)

Here, \( a \) is the first term of the series, and \( r \) is the common ratio.
Applying this to the original exercise:

• The first term \( a = 4 \).
• The common ratio \( r = \frac{y}{3} \).

Plug these values into the formula, we get:

• \( S = \frac{4}{1 - \frac{y}{3}} \)
• Simplifying, \( S = \frac{12}{3-y} \)

This formula gives us the precise sum of the series within its convergence range \(|y| < 3\). It's a straightforward calculation once \( a \) and \( r \) are known.

A **geometric series** is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio \( r \).
The general expression for a geometric series can be written as:

• \( a, ar, ar^2, ar^3, \ldots \)

This series has particular significance due to its simplicity and its well-defined sum formula:

• \( S = \frac{a}{1-r} \), provided \(|r| < 1\)

Recognizing a series as geometric involves identifying a constant factor between consecutive terms. In our exercise, the sequence \( 4 + y + \frac{y^2}{3} + \frac{y^3}{9} + \cdots \) follows this pattern, with each term being a fixed ratio \( \frac{y}{3} \) of the prior term.
This formula is crucial because it provides a method to determine the sum of all series terms, as long as the series converges. By agreeing to use a consistent fixed ratio, the geometric series becomes one of the fundamental concepts in mathematical analysis of infinite sums.