Problem 105 - Precalculus with Limits

Question

In Exercises 93 - 106, find the sum of the infinite geometric series. \( \dfrac{1}{9} - \dfrac{1}{3} + 1 - 3 + \cdots \)

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is \( -0.025 \).

Step 1: Identifying the first term (a)

The first term \( a \) of the geometric series is the first number in the series, which is \( \dfrac{1}{9} \).

Step 2: Identifying the common ratio (r)

The common ratio \( r \) of the geometric series can be obtained by dividing the second term by the first term, or dividing the third term by the second term, and so on. Here, \( r = \dfrac{- \dfrac{1}{3}}{\dfrac{1}{9}} = -3 \).

Step 3: Applying the formula for the sum of an infinite geometric series

Using the formula \( S = \dfrac{a}{1-r} \), we can substitute \( a = \dfrac{1}{9} \) and \( r = -3 \), which gives us \( S = \dfrac{\dfrac{1}{9}}{1-(-3)} = -0.025 \).

Key Concepts

Common Ratio

In a geometric series, the common ratio is a critical component. It determines how each term in the series relates to the one before it. To find the common ratio \( r \), you divide any term in the series by the preceding term. This process needs to be consistent for every consecutive term.

For instance, if you begin with a series like \( \frac{1}{9}, -\frac{1}{3}, 1, -3, \ldots \), you find the common ratio by calculating \( r = \frac{-\frac{1}{3}}{\frac{1}{9}} \). Here, each fraction represents a term, and the division shows how the series transforms from one term to the next. Finding this ratio is crucial because it directs whether or not the series will converge, particularly in infinite series.

Geometric Series Sum Formula

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

With this formula, you can determine the sum of an infinite geometric series provided that the absolute value of \( r \) is less than 1, which ensures the series converges. If \( |r| \geq 1 \), then the series does not have a sum, as it either diverges or equals infinity. In our example, this formula was applied with \( a = \frac{1}{9} \) and \( r = -3 \), which interestingly indicates a divergence rather than a valid sum for a converging series.

First Term Identification

Identifying the first term \( a \) is the foundational step when dealing with any geometric series, including those that are infinite. It is simply the initial value you start from in the sequence of numbers.

For the series \( \frac{1}{9}, -\frac{1}{3}, 1, -3, \ldots \), the first term, \( a \), is \( \frac{1}{9} \). This term sets the stage for the rest of the series because it is the reference point from which the series evolves based on the common ratio. Acknowledging this first term correctly ensures accurate calculations for further analysis, including determining possible convergence and calculating the series sum using the geometric series sum formula.