91Ó°ÊÓ

Infinite geometric series can be a bit intimidating, but understanding their convergence is key to solving them. A series is simply a sum of terms. Each term of a geometric series after the first is found by multiplying the previous term by the common ratio, denoted as \( r \).

The convergence of an infinite geometric series depends on this common ratio. Specifically, if the absolute value of the common ratio \( |r| \) is less than one, the series will converge, meaning its sum will approach a specific finite number as more terms are added.

If \(|r|\) is equal to or greater than one, the series will diverge, tending towards infinity or fluctuating indefinitely without settling to a particular sum.

• A convergent series implies that we can calculate a precise sum using a specific formula if \(|r| < 1\).
• Convergence simplifies the infinite series into something that can be easily managed: a finite number.

Let's dive into the concept of the common ratio, a fundamental aspect of any geometric series. The common ratio \( r \) is the factor by which we multiply each term of the series to obtain the next one. It is a constant that remains the same throughout the entire series.

To find the common ratio, divide any term in the series by its preceding term. For example, given the series \( 1000, -300, 90, \ldots \), you can find \( r \) by dividing the second term \(-300\) by the first term \(1000\). This gives us \( r = -0.3 \).

The sign of the common ratio also affects the series:

• If \( r \) is positive, all terms will be of the same sign.
• Negative \( r \) will alternate terms between positive and negative.

Understanding and calculating the common ratio is essential for determining the behavior of the series.

After identifying that a geometric series converges, you can easily find its sum using the geometric series sum formula. This formula is crucial as it allows you to compute the total sum of an infinitely long series if the series converges.

The formula for the sum \( S \) of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \]where \( a \) is the first term of the series and \( r \) is the common ratio.

In our example, the first term \( a \) is \( 1000 \) and the common ratio \( r \) is \(-0.3\). Substituting these values into the formula gives:

• \( S = \frac{1000}{1 - (-0.3)} = \frac{1000}{1.3} \)
• Simplifying gives an approximate sum of \( 769.23 \).

This calculation demonstrates how a potentially infinite series can be expressed as a finite number, making the series much more manageable for practical use.