Problem 22 - Algebra 2

Question

Evaluate each infinite geometric series. $$ 3+1+\frac{1}{3}+\frac{1}{9}+\ldots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(3 + 1 + \frac{1}{3} + \frac{1}{9} + ...\) is 4.5.

Step 1: Identify the first term (a) and the common ratio (r)

For this series, the first term \(a\) is 3 and the common ratio \(r\) obtained by dividing the second term by the first term, or the third term by the second term and so on is \(\frac{1}{3}\).

Step 2: Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is \(S = \frac{a}{1 - r}\). For this series, \(a = 3\) and \(r = \frac{1}{3}\), so we substitute these values into the formula to find the sum \(S\).

Step 3: Simplify the expression to find the sum

Substituting \(a = 3\) and \(r = \frac{1}{3}\) into the formula, we get \(S = \frac{3}{1 - \frac{1}{3}}\). Simplify the denominator to get \(S = \frac{3}{\frac{2}{3}}\). Then simplify the expression to get \(S = \frac{9}{2}\), or 4.5 in decimal form.

Key Concepts

Common Ratio

In the context of an infinite geometric series, the common ratio is a crucial element that defines the relationship between consecutive terms. Think of it as a multiplier that stays constant throughout the series, connecting each term to the next. For instance, in our example series, the terms start with 3, followed by 1, then \(\frac{1}{3}\), and so on.

• To find the common ratio \(r\), simply divide any term in the series by the previous one. In our case, dividing 1 by 3 gives us the common ratio: \(\frac{1}{3}\).
• The consistency of the common ratio helps in identifying the geometric nature of the series.

Understanding the concept of the common ratio is important because it determines whether the series converges (has a finite sum) or diverges. For a series to converge, the absolute value of the common ratio has to be less than 1, \(|r| < 1\). This ensures the terms get progressively smaller, allowing for a finite sum.

Sum of a Series

The sum of an infinite geometric series might initially seem puzzling. After all, how can an endless series add up to a finite number? This is where the magic of convergence comes into play.

• If the series has a common ratio \(r\) where \(|r| < 1\), the series approaches a certain finite sum as more and more terms are added.

For example, in our series, we begin with 3 and add room for smaller terms through successive multiplication by \(\frac{1}{3}\). This results in a pattern where each term adds less and less to the total sum. By understanding this behavior, students can better grasp why a sequence that never seems to end can still settle at a finite value like 4.5.

Geometric Series Formula

The geometric series formula provides a concise way to find the sum of an infinite geometric series. The formula is expressed as follows: \[ S = \frac{a}{1 - r} \]Where \(a\) is the first term of the series and \(r\) is the common ratio. Applying this formula helps to quickly derive the sum of an otherwise complex series.

• In our exercise, the first term \(a\) is 3, while \(r\) is \(\frac{1}{3}\).
• Substitute these into the formula, \(S = \frac{3}{1 - \frac{1}{3}}\), to get \(S = \frac{3}{\frac{2}{3}}\).

This simplifies to \(S = \frac{9}{2}\), or 4.5 in decimal form. Using the formula not only clarifies the series sum but also provides insight into the efficiency of finding such sums without laborious addition or estimation methods.