Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Find each sum. $$\sum_{k=1}^{\infty}(0.1)^{k}$$

Step-by-Step Solution

Answer

The sum is \( \frac{1}{9} \) or approximately 0.1111...

Step 1: Identify the Sequence

The sum given is an infinite series: \( \sum_{k=1}^{\infty}(0.1)^{k} \). This is a geometric series with each term defined as \( a_k = (0.1)^k \). The first term \( a_1 \) is \( 0.1 \).

Step 2: Confirm Series is Geometric

In a geometric series, each term is obtained by multiplying the previous term by a constant ratio, \( r \). Here, the series is geometric with a common ratio \( r = 0.1 \).

Step 3: Check Condition for Sum of Infinite Series

The sum of an infinite geometric series \( S = \sum_{k=1}^{\infty} ar^k \) exists if the absolute value of the common ratio is less than 1 (\(|r| < 1\)). Here \( |0.1| < 1 \), so the sum exists.

Step 4: Use Infinite Geometric Series Formula

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. In this case, \( a = 0.1 \) and \( r = 0.1 \).

Step 5: Calculate the Sum

Plug the values into the formula: \( S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9} \). Simplify to find the sum: \( S = \frac{1}{9} \approx 0.1111... \) (repeating decimal).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term is derived by multiplying the previous one by a fixed, non-zero number called the "common ratio." This type of series is defined through its simplicity of creating each following term using the same multiplicative constant. An example of a geometric series is:

• First term: \( a_1 \)
• Second term: \( a_2 = a_1 imes r \)
• Third term: \( a_3 = a_2 imes r \)

You can see the pattern forming, thus confirming the sequence's geometric nature. In our exercise, the series given is \( \sum_{k=1}^{\infty}(0.1)^{k} \), making it a geometric series since each term is formed by multiplying the previous one by 0.1. This shows how geometric series can have applications across different mathematical calculations and simplifies computations by following this clear pattern.

Common Ratio

The common ratio, often denoted as \( r \), is a vital part of the geometric series. It is the factor by which each term is multiplied to produce the next term. Essentially, it determines how sharply or gently the series increases or decreases. If \( |r| < 1 \), the series will converge towards a particular value.

• For example, if \( r = 0.5 \), the terms will halve each time, getting smaller quickly.
• If \( r = 2 \), terms will double, increasing rapidly.

In the given exercise, the common ratio is 0.1, indicating that the terms will diminish slowly, making it a convergent series (\({r}\) is positive but less than 1). By understanding and calculating the common ratio, you determine both the nature and behavior of the series involved.

Sum of Series

Calculating the sum of a series, particularly when dealing with infinite series, is an intriguing aspect of geometric series.
For a geometric series, when \( |r| < 1 \), 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 formula collapses an infinite number of terms into a simple expression. For our exercise, with \( a = 0.1 \) and \( r = 0.1 \), substituting the values gives us: \[ S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9} = \frac{1}{9} \approx 0.1111... \]. This efficient method provides the sum of terms stretching out to infinity, showing how structured approaches like this bring order to what initially may seem infinite and unwieldy.

Series Convergence

Series convergence is a fundamental notion when working with infinite series. A series converges when it approaches a finite value as more terms are added. In contrast to that, divergence refers to a series whose sum increases without bound. For a geometric series to converge, the common ratio's absolute value must be less than 1 ( \(|r| < 1\)).

• For example, a series with \( r = 0.5 \) will converge since each term shrinks.
• Conversely, a series with \( r = 2 \) will grow indefinitely, thus diverging.

The problem we're handling in our task employs a common ratio of 0.1. Since \( 0 < |0.1| < 1 \), the series converges. Understanding this concept helps assure that the infinite sequence indeed sums to a finite number, revealing how conditions of convergence lead to meaningful and usable mathematical results.