Problem 17 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The given infinite geometric series diverges and therefore does not have a sum.

Step 1: Identify the first term and the common ratio

First identify the first term of the series which is \(a = 1/4\). Then calculate the common ratio \(r\). For this series, the ratio between one term and the previous term is constant (each term is twice the previous one). Therefore, the common ratio \(r = 2\).

Step 2: Check for convergence

To determine whether the series converges, check if the absolute value of the common ratio is less than one, i.e., \(|r| < 1\). For the given series, the common ratio \(r = 2\) and its absolute value is not less than 1. This means the series diverges.

Step 3: State whether the series has a sum

Since the series diverges, it does not have a sum. An infinite geometric series only has a sum if it converges, i.e., if the absolute value of the common ratio is less than 1.

Key Concepts

Divergence

When we speak about the divergence of an infinite geometric series, we're discussing a series that does not approach a specific value. This means that as we continue to add more terms, the sum gets larger without approaching a limit. One crucial factor that determines divergence is the common ratio. If the absolute value of this ratio is greater than or equal to 1, the series will diverge.
For instance, in the series \[ \frac{1}{4} + \frac{1}{2} + 1 + 2 + \ldots \], the common ratio \( r = 2 \) causes the series to expand exponentially. Here, because \( |r| = 2 \) is greater than 1, the terms keep increasing, leading to a divergent series. A divergent series cannot have a finite sum because it doesn't settle down to a single value.

Convergence

In contrast, an infinite geometric series is said to converge if it approaches a specific finite value as more terms are added. Convergence can only occur if the common ratio's absolute value is less than 1.
This is because when the common ratio \( |r| < 1 \), each term becomes successively smaller, allowing the sum to stabilize toward a particular number.

• For example, if we had a series with a common ratio of \( r = \frac{1}{2} \), each subsequent term would be half the size of the previous one, gradually reducing.
• This reduction means the total sum of the series can approach a fixed limit.

If a series converges, we can calculate its sum using a specific formula, but only when \( |r| < 1 \). Unfortunately, our given series does not allow for convergence due to its common ratio exceeding this specified range.

Common Ratio

The common ratio is fundamental in determining the behavior of a geometric series. It is the factor by which we multiply a term to get the next term in the series. Knowing the common ratio helps us predict whether a series will converge or diverge.
In our example, the common ratio \( r = 2 \) was identified by noting that each term is twice as large as the one before.

• If the absolute value of the common ratio is less than 1, the series converges.
• If it is equal to or greater than 1, as with our series, the series diverges.

Identifying the common ratio is central to solving and understanding infinite geometric series, allowing us to quickly determine the nature of the series without calculating endless terms.