Problem 14 - Algebra 2

Question

Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. $$ -54-18-6-\dots $$

Step-by-Step Solution

Answer

The series converges and its sum is -81.

Step 1: Identify the common ratio

In this geometric series, the common ratio (r) can be calculated by dividing the second term by the first term. So, r = \(-18/-54 = 1/3\). Similarly, check if the ratio is the same for the second and third term. It is decided that the series is geometric with ratio 1/3.

Step 2: Check convergence or divergence

An infinite geometric series converges if the absolute value of the common ratio r is less than 1 and diverges otherwise. Since the absolute value of r (1/3) is less than 1, therefore it is converging.

Step 3: Find the sum

The sum of an infinite geometric series is given by the formula, \[S = a / (1 - r)\] where S is the sum, a is the first term, and r is the common ratio. Plugging the values a = -54 and r = 1/3 into the formula, we get the sum S = \(-54 / (1 - 1/3) = -81\).

Key Concepts

Convergence and Divergence of Series

In the world of infinite series, understanding whether a series converges or diverges is crucial. Convergence means the series approaches a definite number as you sum its terms to infinity. Divergence implies the series grows without bound or oscillates indefinitely.
For geometric series, the key factor is the "common ratio" (denoted as \( r \)).
A geometric series converges if the absolute value of its common ratio \( |r| \) is less than 1.
If \( |r| \geq 1 \), the series diverges.It's useful to remember:

• If a series converges, it has a sum.
• If it diverges, it does not have a sum.

This information helps you determine the behavior of the series and find its sum if it converges.

Infinite Series

An infinite series is a sum of an infinite sequence of numbers. The series given here, \(-54 - 18 - 6 - \dots\), is an example of an infinite geometric series.
Infinite series can either converge or diverge depending upon the series terms' arrangement.
In geometric series, the construction is simple because each term is a product of the previous term and a constant common ratio, \( r \).
This predictable pattern helps analyze such series effectively.
By understanding whether a series converges or diverges, you can assess if it is meaningful to calculate its sum.

Common Ratio

The common ratio is a core concept when dealing with geometric series. It is the factor by which we multiply each term to get the next term in the series.
To find the common ratio \( r \), simply divide any term by its preceding term.
In the series \(-54, -18, -6, \dots\), \( r = \frac{-18}{-54} = \frac{1}{3} \).
This ratio confirms the geometric nature of the series.

A few important points:

• A positive \( r \) keeps terms in the same sign direction.
• A negative \( r \) flips the sign of terms each time.
• If \( |r| < 1 \), terms decrease in magnitude, aiding convergence.

This simple calculation can tell you a lot about the series behavior.

Sum of a Series

For convergent geometric series, finding the sum is straightforward with a formula. The sum \( S \) of a converging infinite geometric series is calculated as:\[ S = \frac{a}{1 - r} \]where \( a \) is the first term and \( r \) is the common ratio.
In our example, \( a = -54 \) and \( r = \frac{1}{3} \), so:\[ S = \frac{-54}{1 - \frac{1}{3}} = -81 \]This tells us that the series totals to \(-81\) if summed to infinity.
Remember, this sum formula is only applicable when the series converges (i.e., \(|r| < 1\)).

Being able to find the sum of a convergent series allows you to quantify its total value comprehensively.