Problem 45 - College Algebra

Question

Find the sum of the infinite geometric series. $$ \sum_{n=1}^{\infty} 4.6 \cdot 0.5^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is 9.2.

Step 1: Identify the First Term

To find the sum of the infinite geometric series, we first identify the first term, \( a \), which is the term when \( n=1 \). Substituting \( n=1 \) into the expression gives \( 4.6 \cdot 0.5^{1-1} = 4.6 \cdot 0.5^0 = 4.6 \cdot 1 = 4.6 \). Thus, \( a = 4.6 \).

Step 2: Identify the Common Ratio

The common ratio \( r \) is obtained from the exponential part of the series: \( 0.5^{n-1} \). The base of the exponent \( 0.5 \) is the common ratio. Hence, \( r = 0.5 \).

Step 3: Check Convergence Condition

An infinite geometric series converges when the absolute value of the common ratio \( |r| \) is less than 1. Since \( |0.5| = 0.5 < 1 \), the series converges.

Step 4: Use Infinite Series Sum Formula

The sum \( S \) of an infinite geometric series is given by the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term, and \( r \) is the common ratio. Substituting the given values \( a = 4.6 \) and \( r = 0.5 \), the formula becomes \( S = \frac{4.6}{1 - 0.5} = \frac{4.6}{0.5} \).

Step 5: Calculate the Sum

Calculate \( S = \frac{4.6}{0.5} = 4.6 \times 2 = 9.2 \). Thus, the sum of the infinite geometric series is 9.2.

Key Concepts

First Term

In an infinite geometric series, identifying the "first term" is a crucial step. This first term is generally denoted by the letter \( a \). It represents the first number in your series sequence. Here's how it works: when you have the function for the series, plug in \( n = 1 \) because that tells you what the first term is.

In our exercise, the function for the series is \( 4.6 \cdot 0.5^{n-1} \). So when \( n = 1 \), this becomes \( 4.6 \cdot 0.5^{1-1} = 4.6 \cdot 0.5^0 = 4.6 \cdot 1 = 4.6 \).

Therefore, our first term \( a = 4.6 \). Understanding the first term is your starting point for any calculations in a geometric series. It's like knowing where the journey begins!

Common Ratio

The term "common ratio" in an infinite geometric series is very important because it defines how each term relates to the previous one. Let's think of the common ratio as the factor by which you multiply one term to get the next term in the series.

In mathematical terms, for our series, the base of the exponent in \( 0.5^{n-1} \) is what tells us the common ratio. So here, the base \( 0.5 \) is the common ratio \( r \). Every term in the series is obtained by multiplying the previous term by this constant factor \( r \).

A quick example can help. If the first term is \( 4.6 \), the second term would be \( 4.6 \times 0.5 \), the third \( 4.6 \times 0.5 \times 0.5 \), and so on. This ratio remains consistent throughout the series.

Convergence Condition

The convergence condition is what decides if an infinite geometric series will actually have a finite sum. Simply put, a series can only converge to a finite sum if its common ratio \( r \) satisfies this condition: the absolute value \( |r| \) must be less than 1.

This makes sense because if \( |r| \) were to be 1 or greater, the terms in the series could keep getting larger, leading to an infinite total, which is not useful if you want a specific sum.

Let's apply this to our example. Here the common ratio \( r = 0.5 \), and \( |0.5| = 0.5 \), which is indeed less than 1, so our series meets the convergence condition. This means that the series shrinks down and totals to a finite number.

Sum Formula

Once we're sure that the series converges, we can use the sum formula to find its total. The sum formula for an infinite geometric series is given by:

• \( S = \frac{a}{1-r} \)

Here, \( S \) is the sum we want to find, \( a \) is the first term, and \( r \) is the common ratio from our series.

In our calculation, we substitute \( a = 4.6 \) and \( r = 0.5 \) into the formula:

• \( S = \frac{4.6}{1 - 0.5} = \frac{4.6}{0.5} \)

By solving this, we get \( S = 4.6 \times 2 = 9.2 \).

So the sum of our infinite geometric series is 9.2. This is how you apply the formula to complete the calculation!