Problem 118 - Precalculus with Limits

Question

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{k=1}^{\infty} \left(\frac{1}{10} \right)^k \)

Step-by-Step Solution

Answer

The sum of the infinite series is \( S = \frac{1}{9} \)

Step 1: Identify the first term and the common ratio

From the series \( \displaystyle \sum_{k=1}^{\infty} \left(\frac{1}{10} \right)^k \), it can be seen that the first term \(a\) is \( \left( \frac{1}{10} \right)^1 = \frac{1}{10} \). The common ratio \(r\) is \( \frac{1}{10} \).

Step 2: Check the condition for the common ratio

The geometric series converges when \( |r| < 1 \), which is \(-1< r < 1\). Here, \(r = \frac{1}{10}\) and it satisfies this condition.

Step 3: Calculate the sum of the infinite series

Apply the formula for the sum of an infinite geometric series, \( S = \frac{a}{1 - r}\). Substituting \(a = \frac{1}{10}\) and \(r = \frac{1}{10}\) we have, \( S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} \). Simplify this to find the sum.

Key Concepts

Convergence of Series

Understanding the convergence of series is crucial when dealing with infinite series. A series converges if the sum of its terms approaches a specific value as the number of terms increases indefinitely. Conversely, a series diverges if the terms do not approach any specific value, no matter how many terms are added.

A simple way to think about convergence is using a real-life analogy: imagine filling a jar with water droplets. If the jar will eventually fill up at some predictable level, we can say the 'droplet series' converges. If the jar has no bottom and the water just keeps flowing through, this would be like a divergent series – there's no accumulation to a definitive level.

In mathematical terms, an infinite series like \[\begin{equation} \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \text{\textellipsis} \end{equation}\] is said to be convergent if the sum of all its infinite terms can be summed up to a finite number. To determine this, we need to analyze the series' behavior and apply certain tests – one fundamental test for a geometric series is the comparison of its common ratio to 1 in absolute terms. If the absolute value of the common ratio is less than 1, we say the series is convergent.

Geometric Series

A geometric series is a series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. You can recognize a geometric series by the fact that the ratio between consecutive terms is always the same.

For instance, in the series \[\begin{equation} \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \text{\textellipsis}, \end{equation}\] you can see that each term is \[\begin{equation} \frac{1}{10} \end{equation}\] times the term before it. This constant factor is what defines the series as geometric. We focus on calculating sums of such series because they frequently appear in various areas like finance, computer science, and physics.

For an infinite geometric series to have a finite sum, as mentioned previously, the common ratio must have an absolute value less than 1, leading to the concept of convergence. In a finite geometric series or a convergent infinite geometric series, you can compute the sum using specific formulas, which significantly simplifies the analysis and calculations of such series.

Common Ratio

The common ratio of a geometric series is a central element in understanding its behavior. It is denoted as 'r' and represents the factor by which we multiply a term to get the next term in the series. The sign and magnitude of the common ratio have important implications on the series' nature.

For example, when the common ratio is between -1 and 1 (but not equal to zero), the series can converge as your terms get smaller (approaching zero). In contrast, if the common ratio is greater than 1 or less than -1, each term grows larger in magnitude, and the series diverges.

How to Identify the Common Ratio

You can find the common ratio by taking any term in the series (after the very first one) and dividing it by the previous term. In the provided exercise, the common ratio is \[\begin{equation} r = \frac{\frac{1}{10^n}}{\frac{1}{10^{n-1}}} = \frac{1}{10}, \end{equation}\] which clearly shows a consistent division by 10. Once you know the common ratio, you can use it to determine the convergence of the series and to find the sum if the series is indeed convergent.