Question

For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. $$0.6+0.06+0.006+\cdots$$

Short answer

Answer: The conjectured value of the infinite series is $\frac{2}{3}$.

Step-by-step solution

Partial Sum 1

The first partial sum is just the first term of the series: $$S_1 = 0.6$$

Partial Sum 2

To find the second partial sum, we add the second term to the first partial sum: $$S_2 = 0.6 + 0.06 = 0.66$$

Partial Sum 3

Similarly, to find the third partial sum, we add the third term to the second partial sum: $$S_3 = 0.66 + 0.006 = 0.666$$

Partial Sum 4

Finally, for the fourth partial sum, we add the fourth term to the third partial sum: $$S_4 = 0.666 + 0.0006 = 0.6666$$

Conjecture about the value of the infinite series

Based on the behavior of the partial sums, it seems that as we keep adding more terms to the series, the value of the sum is approaching 2/3 or 0.666... (with an infinite string of 6's). Thus, our conjecture for the value of the infinite series is: $$\sum_{n=1}^{\infty} 0.6(0.1)^{n-1} = \frac{2}{3}$$

Key concepts

Partial Sums

Understanding the concept of partial sums is essential when dealing with infinite series. A partial sum is simply the sum of the first n terms of a sequence. In the context of our example, we're looking at the infinite series with terms 0.6, 0.06, 0.006, and so on, where each term is ten times smaller than the previous one.

To grasp what a partial sum represents, it's helpful to think of it as a snapshot in the process of adding up the terms of an infinite series. For instance, if we were to stop after the first term, the partial sum, denoted as \( S_1 \), would be 0.6. If we include the second term, our new partial sum, \( S_2 \), is 0.6 + 0.06, which equals 0.66. This process can continue for any number of terms.

Calculating the first four partial sums of the given series, we get 0.6 for \( S_1 \), 0.66 for \( S_2 \), 0.666 for \( S_3 \), and 0.6666 for \( S_4 \). As we add more terms, the partial sums grow closer to a certain value, which leads us to investigate the convergence of the series.

Geometric Series

The series in our exercise is a type of infinite series known as a geometric series. Geometric series have a common ratio between consecutive terms. For the series 0.6, 0.06, 0.006, ..., each term is obtained by multiplying the previous term by 0.1, making the common ratio 0.1.

A standard geometric series is written as \( a, ar, ar^2, ar^3, ... \), where \( a \) is the first term and \( r \) is the common ratio. These series are significant because they have a known formula for their sum, which is \( S = \frac{a}{1 - r} \), provided that the absolute value of \( r \) is less than one (\( |r| < 1 \)). This condition ensures that the terms get smaller and the series converges, which leads to our next concept of convergence.

In our specific case, the first term \( a = 0.6 \) and the common ratio \( r = 0.1 \). By using the geometric series sum formula, we get the sum of the series as \( S = \frac{0.6}{1 - 0.1} = \frac{2}{3} \), just like our conjecture from the partial sums analysis suggested.

Convergence of Series

The convergence of an infinite series is a fundamental concept in mathematics that determines whether the series has a finite limit as the number of terms approaches infinity. If an infinite series converges, the total sum can be calculated using various techniques, depending on the type of series we are dealing with.

For a geometric series, convergence is determined by the common ratio \( r \). If \( |r| < 1 \), the series converges to the sum \( S = \frac{a}{1 - r} \). Otherwise, if the ratio is greater than or equal to one in absolute value, the series does not converge, and the sum is considered infinite.

Applying this to our example, since the common ratio \( r = 0.1 \) is less than one, the series is convergent. Consequently, the infinite series sum gradually approaches, or 'settles down', to the value \( \frac{2}{3} \) as we compute more and more partial sums. This is a perfect illustration of how the behavior of partial sums can lead to an understanding of the convergence and overall behavior of an infinite series.