Problem 39 - Algebra and Trigonometry

Question

Find the sum of each infinite geometric series. $$ 3+\frac{3}{4}+\frac{3}{4^{2}}+\frac{3}{4^{3}}+\cdots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is 4.

Step 1: Identify the First Term

The first term of an infinite geometric series usually is the first given term. The first given term is 3, so \(a = 3\).

Step 2: Identify the Common Ratio

The common ratio of the sequence is the ratio of the second term to the first term. Let's take the second term which is \(\frac{3}{4}\) and divide by the first term which is 3. This results in \(\frac{\frac{3}{4}}{3}=\frac{1}{4}\). Therefore, \(r = \frac{1}{4}\).

Step 3: Calculate the Sum

Using the formula \(S=\frac{a}{1-r}\) for the sum of an infinite geometric series, replace \(a\) with 3 and \(r\) with \(\frac{1}{4}\) to get \(S=\frac{3}{1-\frac{1}{4}}\). Simplify this to find that \(S=4\).

Key Concepts

Sum of a Geometric Series

Understanding the sum of an infinite geometric series can be quite fascinating. A geometric series is made up of terms that are produced by multiplying the previous term by a constant. This constant is known as the common ratio. Imagine placing dominos in a line, each a bit further than the last - that's analogous to the terms in a geometric series growing larger or shrinking, based on the common ratio.

Now, for an infinite series to have a finite sum, the terms need to get smaller; in other words, the common ratio should be between -1 and 1. The formula to sum an infinite geometric series is given by: \[ S=\frac{a}{1-r} \] where \( S \) represents the sum, \( a \) is the first term, and \( r \) is the common ratio. Using this magical key, if a series does converge, we can unlock the total of all its infinite terms in a single number. For our exercise, with \( a = 3 \) and \( r = \frac{1}{4} \), the sum beautifully collapses into the number 4.

Common Ratio

The common ratio in a geometric series is like the DNA—every term is formed by multiplying the previous term by this factor. Identifying it is essential as it determines if the series converges or diverges. While it is the constant multiplier from one term to the next in a sequence, its value offers a peek into the behavior of the series over time.

When the common ratio \( r \) is less than 1 in absolute value, each term in the series gets smaller, gently approaching zero. If the common ratio is greater than 1 in absolute value, the terms grow without bounds—head towards infinity. In our example, we determined the common ratio to be \( \frac{1}{4} \), which is less than 1, signifying that the series converges and that we can indeed find its sum.

Convergence of Series

The convergence of an infinite series is a term that describes whether the series approaches a finite value as the number of terms goes towards infinity. Think of it like shooting arrows at a target. If your arrows land closer and closer to the bull's-eye, your aim (or series) converges. However, if your arrows keep hitting further away, you're diverging from the target, just like a divergent series.

In the realm of infinite geometric series, convergence is determined by that all-important common ratio. Specifically, when \( |r| < 1 \), the series converges. This means as you add more terms, they add up to a particular number, rather than spiraling off to infinity. In our exercise, the ratio was \( \frac{1}{4} \), so our infinite sum found its target—convergence! We calculated the sum to be 4, illustrating that infinite does not always mean immeasurable.