Problem 51 - Algebra 2

Question

Determine whether the sum of each infinite geometric series exists. $$ 4+2+1+\frac{1}{2}+\frac{1}{4}+\ldots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is 8.

Step 1: Find the common ratio

The common ratio, \( r \), in a geometric series is found by dividing any term by its preceding term. Here, from the provided series it is possible to take 2 (second term) divided by 4 (first term), which equals to 0.5. So, \( r = 0.5 \).

Step 2: Determine the convergence of the series

A geometric series \( a+ar+ar^2+ar^3+... \) converges if -1 < r < 1. In this case, the common ratio (\( r = 0.5 \)) is between -1 and 1, which means the series converges.

Step 3: Find the sum of the series

If the geometric series converges, we can find its sum using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. Substituting the given values: \( a = 4 \) and \( r = 0.5 \) we get \( S = \frac{4}{1-0.5} = 8 \).

Key Concepts

Geometric Series Convergence

In the world of mathematics, an infinite geometric series can be quite intriguing. This type of series can continue indefinitely, yet under certain conditions, it converges to a specific value. To determine whether an infinite geometric series converges, we primarily need to consider the common ratio, denoted as \( r \). If the series is expressed in the form \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio, convergence occurs when \(-1 < r < 1\). For the series \( 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \ldots \), as calculated, our \( r = 0.5 \), which falls within this range. Thus, we can confidently say this series converges.

Common Ratio

A vital part of identifying whether a geometric series will converge is determining its common ratio (\( r \)). The common ratio is found by dividing any term in the series by its preceding term. Let's take a closer look at how this works. For the series given, \( 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \ldots \), you can calculate the ratio by dividing 2 by 4. Just like this: \( \frac{2}{4} = 0.5 \). The same is true for other pairs of successive terms such as \( \frac{1}{2} \div 1 = 0.5 \). Repeating this process confirms that \( r = 0.5 \). This consistency assures us of our common ratio, an important factor in further calculations.

Geometric Series Sum

Once we've established that a geometric series converges, finding its sum becomes the next fascinating step. The sum \( S \) of an infinite geometric series can be calculated using the simple formula:

• \( S = \frac{a}{1-r} \)

Here, \( a \) represents the first term of the series, and \( r \) the common ratio. Applying this to our sequence:

• First term \( a = 4 \)
• Common ratio \( r = 0.5 \)

Plug these values into the formula to get:

• \( S = \frac{4}{1-0.5} \)
• \( S = \frac{4}{0.5} \)
• \( S = 8 \)

Thus, the sum of this infinite series is 8, meaning that as the terms endlessly progress, they collectively total to 8.

Convergence Criteria

Understanding the convergence of an infinite geometric series is hinged on specific criteria related to its common ratio \( r \). These criteria guide us in predicting whether a series will yield a finite sum without unraveling forever. The criteria state:

• The series converges if \(-1 < r < 1\).
• Conversely, if \( |r| \geq 1 \), the series diverges, implying no finite sum can be achieved.

In our example, \( r = 0.5 \), which satisfies \(-1 < r < 1\). Therefore, the criteria are met, and the convergence is assured. This understanding helps in many practical applications where summing an infinite series is necessary, such as in calculating present values in finance or analyzing signals in engineering. It provides a clear path to ensuring mathematical series are not only meaningful but also manageable.