Problem 11 - Algebra 2

Question

Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. $$ 4+2+1+\ldots $$

Step-by-Step Solution

Answer

The given infinite geometric series converges and has a sum of 8.

Step 1: Identify the series as geometric

A geometric series is a series with a common ratio between each term. In this case, each term is exactly half of the preceding term, hence the common ratio \( r \) for this series is 0.5.

Step 2: Determine if the series converges

An infinite geometric series converges if the absolute value of the common ratio \( r \) is less than 1. For this series, since \( |0.5| < 1 \), the series converges.

Step 3: Calculate the sum

For a convergent geometric series, the sum can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 4 \) and \( r = 0.5 \), so the sum \( S = \frac{4}{1 - 0.5} = 8 \)

Key Concepts

Convergence and Divergence of Infinite Geometric Series

When dealing with infinite geometric series, the terms "convergence" and "divergence" are key to understanding if the series has a finite sum. Convergence occurs when the infinite series approaches a specific value, while divergence happens when the series increases indefinitely without approaching any particular point.

In mathematical terms, an infinite geometric series converges if the absolute value of the common ratio, denoted as \( r \), is less than 1. That means \( |r| < 1 \). If this condition is met, each successive term gets smaller and adds less to the overall sum, allowing the series to converge on a specific limit.

• If \( |r| < 1 \), the series converges, and a sum can be calculated.
• If \( |r| \geq 1 \), the series diverges, meaning no finite sum exists.

In the problem at hand, the infinite series with terms \( 4 + 2 + 1 + \ldots \) has a common ratio \( r = 0.5 \), so it converges. Hence, it has a finite sum.

Common Ratio in Geometric Series

The common ratio is a constant factor that we multiply each term by to obtain the next term in a geometric series. It's a central element that dictates whether the series converges or diverges. The common ratio \( r \) is found by dividing any term in the series by the previous term.

For the series given in the exercise, starting with 4 and continuing as 2, 1, and so forth, we determine \( r \) by dividing each term by its predecessor, i.e., \( \frac{2}{4} = 0.5 \) and \( \frac{1}{2} = 0.5 \). Thus, our common ratio \( r = 0.5 \).

• The value of \( r \) tells us whether the series will converge or diverge.
• A positive \( r \) less than 1 indicates convergence.
• If \( r \) is 0 or approaches 0, this also confirms convergence as terms tend toward zero.

Recognizing the common ratio is essential to any exploration of geometric series, and in this instance, that exploration leads to the conclusion that the series converges.

Sum of a Geometric Series

Calculating the sum of a convergent infinite geometric series can be straightforward when certain conditions are met. The formula \( S = \frac{a}{1 - r} \) is specifically used for series that converge, where \( a \) represents the first term and \( r \) the common ratio.

Given in the problem, our first term is 4, and common ratio \( r \) is 0.5. When we plug into the formula:
\[ S = \frac{4}{1 - 0.5} = \frac{4}{0.5} = 8 \]
This tells us that the infinite series, despite its endless nature, converges to a sum of 8.

• This formula cannot be applied if the series diverges.
• It's crucial for understanding that while individual terms become infinitely small, their sum bounds to 8.

Such calculations show how understanding the interplay between first terms and the common ratio enables us to find meaningful results in infinite series.