Problem 15 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The infinite geometric series \(1-1+1-\ldots\) diverges and thus does not have a sum.

Step 1 Identify the common ratio

The first term, \( a \), is 1. Notice that the series alternates between addition and subtraction of 1. So, the second term is -1 and the common ratio, \( r \), is -1 divided by 1 which equals -1.

Step 2 Decide if the series converges or diverges

The infinite geometric series converges if the absolute value of \( r \) is less than 1, i.e., \( |-1| < 1 \). Clearly, the absolute value of -1 equals to 1, not less than 1, hence the series diverges.

Step 3 State if the series has a sum

Since the series diverges, it does not have a sum. The sum of an infinite geometric series is only defined if the series converges.

Key Concepts

Convergence and Divergence

When we talk about an infinite geometric series, one of the critical considerations is whether the series converges or diverges. Let's break this down. A series is said to converge if as you add more and more terms, you get closer and closer to a specific number. This is like taking very small steps toward a target and eventually landing right on it, even if it takes an infinite number of steps to get there. Conversely, a divergent series does not approach a particular limit. Instead, as you add more terms, the sum keeps getting farther and farther without ever settling down.

For an infinite geometric series, convergence is determined by looking at the common ratio, "r." If the absolute value of "r" is less than 1, the series converges. This means the terms are getting smaller, leading the sum to stabilize at a certain value as more terms are added. In our example, the common ratio was -1, and since the absolute value of -1 is 1, which is not less than 1, the series diverges. So, it continues forever without approaching a definite value.

Common Ratio

Understanding the common ratio is essential when dealing with an infinite geometric series. The common ratio "r" is a constant value that you multiply by each term to get the next term in the series. It's like a glue binding the series together, dictating how each term relates to its neighbors.

To find the common ratio, you take any term (except the first) and divide it by the one that came before it. In our example, the series alternated between 1 and -1. The ratio "r" was therefore calculated as -1 divided by 1, resulting in -1.

This common ratio then plays a vital role in determining whether the series will eventually decelerate to a sum or spiral out of control. A critical property of the common ratio is its absolute value. If |r| < 1, the series tames itself and converges. But if |r| ≥ 1, like in our case where |r| = 1, the series refuses to settle and diverges instead.

Sum of Series

In the world of infinite geometric series, the sum of the series is only meaningful if the series converges. Think of it as a finish line—if the series converges, it crosses the finish line and achieves a sum. But if it diverges, it never settles enough to receive a score.

For a converging 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. This elegant formula gives you the exact value that the entire infinite series approaches. However, if the series diverges—as was the case in our example—this formula doesn't apply. No sum can exist because the series does not approach any particular number. Hence, for our series, because it diverged, we cannot compute a meaningful sum.