Problem 59 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\sum_{k=1}^{\infty}(0.3)^{k}$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{3}{7} \).

Step 1: Identify the Series Type

The series given is \( \sum_{k=1}^{\infty} (0.3)^k \). This is a geometric series with the first term \( a = 0.3 \) and the common ratio \( r = 0.3 \).

Step 2: Check the Convergence Condition

A geometric series \( \sum_{k=1}^{\infty} ar^{k-1} \) converges if the absolute value of the common ratio \( |r| < 1 \). Since \( |0.3| < 1 \), the series converges.

Step 3: Use the Geometric Series Sum Formula

For a convergent geometric series, the sum is given by the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. Plugging in the values, we have \( a = 0.3 \) and \( r = 0.3 \).

Step 4: Calculate the Sum

Substitute the values into the formula: \[S = \frac{0.3}{1-0.3} = \frac{0.3}{0.7} = \frac{3}{7}\]Thus, the sum of the series is \( \frac{3}{7} \).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric series can be either finite or infinite.
For example, in the infinite series \( \sum_{k=1}^{\infty} (0.3)^k \), each term is obtained by multiplying the previous term by \( 0.3 \).
In this series:

• The first term \( a \) is \( 0.3 \).
• The common ratio \( r \) is \( 0.3 \).

The behavior of a geometric series strongly depends on the value of the common ratio \( r \). This influences whether the series converges to a sum or diverges.

Convergence Condition

For an infinite geometric series to converge (i.e., have a finite sum), the absolute value of the common ratio \( r \) must be less than 1. This convergence condition can be written mathematically as \( |r| < 1 \).
Why is this important? Because it tells us that if \( r \) is too large or negative and falls outside this range, the values in the series will not settle to a limit but instead grow indefinitely.

• In our example, the common ratio \( r = 0.3 \), and indeed, \( |0.3| < 1 \). This meets the convergence condition.
• Therefore, we can confidently conclude that the series \( \sum_{k=1}^{\infty} (0.3)^k \) will converge to a finite sum.

Checking the convergence condition is a crucial step when analyzing any geometric series.

Series Sum Formula

Once you establish that a geometric series converges, you can calculate its sum using the series sum formula. The formula for the sum of an infinite convergent geometric series is given by:\[ S = \frac{a}{1 - r} \]
Here, \( a \) is the first term, and \( r \) is the common ratio.
Applying this to our series, where \( a = 0.3 \) and \( r = 0.3 \), we find:

• Substitute these values into the formula: \( S = \frac{0.3}{1 - 0.3} \).
• This simplifies to: \( S = \frac{0.3}{0.7} \).
• On further simplification, the sum of the series \( S \) is \( \frac{3}{7} \).

This formula is a powerful tool for quickly finding the sums of infinite geometric series that meet the convergence condition.