Problem 9 - Algebra 2

Question

Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. $$ 1+\frac{1}{4}+\frac{1}{16}+\ldots $$

Step-by-Step Solution

Answer

The given infinite geometric series converges and its sum is \( \frac{4}{3}\).

Step 1: Identify the First Term and the Ratio

In this geometric series, the first term a is 1 and the ratio r, which is obtained by dividing the second term by the first term, is \(\frac{1}{4}\).

Step 2: Check the Condition for Convergence

For a geometric series to converge, the absolute value of the ratio |r| should be less than 1. Here, |r| = \(\left|\frac{1}{4}\right| < 1\). Therefore, the series converges.

Step 3: Calculate the Sum of the Series

Now that we know the series converges, we can calculate its sum using the formula \(S = \frac{a}{1 - r}\). Substituting a = 1 and r = \(\frac{1}{4}\) into the formula, we get \(S = \frac{1}{1 - 1/4} = \frac{4}{3}\). So, the sum of the series is \( \frac{4}{3}\).

Key Concepts

Convergence and Divergence

When we talk about convergence and divergence in the context of an infinite geometric series, we are asking whether the series reaches a specific sum or not as the number of terms grows infinitely large. This is an essential part of understanding infinite series in mathematics.

To determine this, we look at the geometric ratio, often denoted by \( r \). If the absolute value of this ratio, \( |r| \), is less than 1, the series is said to converge. This means that as you keep adding more terms, the total sum gets closer and closer to a specific number. On the other hand, if \( |r| \) is equal to or greater than 1, the series diverges, indicating that the sum grows without bound or oscillates and never settles at a particular value.

For example, in the series \( 1 + \frac{1}{4} + \frac{1}{16} + \ldots \), the ratio \( r = \frac{1}{4} \) has an absolute value less than 1, hence, it converges. Understanding this concept is crucial as it helps in knowing when a series can be summed to a finite value and when it cannot.

Sum of Series

Once we determine that a geometric series converges, the next step is finding its sum. This is made possible by the formula specifically for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \). Here, \( a \) is the first term and \( r \) is the ratio of the series.

For our example, the series starts with \( a = 1 \) and has a ratio of \( r = \frac{1}{4} \). Plugging these values into the formula gives us:

• \( S = \frac{1}{1 - \frac{1}{4}} \)
• \( S = \frac{1}{\frac{3}{4}} \)
• \( S = \frac{4}{3} \)

This result means as more and more terms are added, the sum approaches \( \frac{4}{3} \).

Recognizing how to use this formula is essential for calculating the sum of any converging geometric series, which can appear in various mathematical and real-world applications.

Geometric Ratio

The geometric ratio \( r \) is a fundamental part of any geometric series. It helps in identifying and determining the nature of the series, whether it converges or diverges.

To find the geometric ratio, you can simply pick any two consecutive terms from the series. Divide one term by the preceding term. The result is your ratio \( r \). In mathematical terms, if your series starts like \( a, ar, ar^2, ar^3, \ldots \), the ratio \( r \) remains the same between each term.

For instance, in the series \( 1 + \frac{1}{4} + \frac{1}{16} + \ldots \), dividing the second term \( \frac{1}{4} \) by the first term 1 gives you \( r = \frac{1}{4} \).

The size of this ratio plays a crucial role in determining if the series will settle to a finite sum (converge) or not (diverge). It acts as a yardstick for unlimited growth or achieving a neat, tidy outcome in an infinite series. This concept is not just theoretical but has practical implications in fields like finance, physics, and computer science.