Problem 39 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 3(0.5)^{n-1}$$

Step-by-Step Solution

Answer

The sum of the series is 6.

Step 1: Identify the first term

The formula for the sum of an infinite geometric series is given by \( S = \frac{a}{1 - r} \) where \( a \) is the first term and \( r \) is the common ratio. Here, the series is \( \sum_{n=1}^{\infty} 3(0.5)^{n-1} \). The first term, \( a \), is the term when \( n = 1 \). This gives us \( a = 3(0.5)^{1-1} = 3 \times 1 = 3 \).

Step 2: Determine the common ratio

The common ratio \( r \) in a geometric sequence is the factor that each term is multiplied by to get the next term. From the series expression \( 3(0.5)^{n-1} \), the base \( 0.5 \) is the common ratio, so \( r = 0.5 \).

Step 3: Check the convergence condition

A geometric series converges if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \). Here, \( |0.5| = 0.5 < 1 \), so the series converges.

Step 4: Apply the sum formula for infinite series

Since the series converges, we can find its sum using the formula \( S = \frac{a}{1 - r} \). Substitute \( a = 3 \) and \( r = 0.5 \) into the formula: \[ S = \frac{3}{1 - 0.5} = \frac{3}{0.5} = 6 \].

Key Concepts

Understanding Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
This specific number sequence is crucial because it creates a pattern that is predictable and disciplined, characterized by consistent multiplication rather than addition.

For example, in the sequence 3, 6, 12, 24, ..., we observe that each term is obtained by multiplying the previous term by 2 (the common ratio here is 2).

• This makes predicting future terms simple once the pattern is known.
• Unlike arithmetic sequences, where we repeatedly add a fixed number, geometric sequences focus on multiplication.

Geometric sequences can grow rapidly if the common ratio is greater than one, or they can shrink towards zero when the common ratio is between zero and one.

The Infinite Sum Formula

The sum of an infinite geometric series can be calculated if certain conditions are met. Specifically, the series must converge, which means its terms get closer and converge to a particular value as more terms are added.
The formula for the sum of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \]

Here:

• \( S \) represents the sum of the series.
• \( a \) is the first term of the series.
• \( r \) is the common ratio.

Using this formula is straightforward once you identify the first term and the common ratio.
Remember, it is only applicable when the absolute value of the common ratio is less than one. Otherwise, the series does not have a finite sum.

Convergence Condition Explained

The convergence condition tells us whether an infinite geometric series has a sum that approaches a finite number. The key condition is that the absolute value of the common ratio must be less than one, represented as: \[ |r| < 1 \]

This condition ensures that the terms of the series become progressively smaller, ultimately allowing the sum to converge to a specific value.

• When \(|r| < 1\), the terms shrink, making the series converge.
• If \(|r| \geq 1\), the terms do not diminish, causing the series to diverge and thus, it doesn’t have a sum.

Checking this condition is a fundamental step before applying the sum formula for infinite series.

The Role of Common Ratio

In a geometric sequence, the common ratio is the multiplier that links each term to the next. It's a central element because it governs the behavior and form of the sequence.
To find the common ratio, divide any term in the sequence by its preceding term.

For example, in the sequence 3, 1.5, 0.75, ..., calculate:

• \( r = \frac{1.5}{3} = 0.5 \)

The common ratio not only determines the growth or decay rate of a sequence but also is vital in determining if the infinite series converges.
If \(|r| < 1\), the sequence has diminishing terms, causing the series to converge. Thus, by understanding and calculating \(r\), we can readily predict the entire behavior and outcome of the sequence’s sum.