Problem 85 - Essential Calculus

Question

In Exercises 85 and \(86,\) (a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1+x+x^{2}+x^{3}+\cdots $$

Step-by-Step Solution

Answer

The common ratio of the geometric series is \(x\), the function that gives the sum of the series is \(S = 1 / (1 - x)\), and graphing this function along with the partial sums \(S_3\) and \(S_5\) shows that as more terms are added to the series, the partial sums get closer to the value of the function \(S = 1 / (1 - x)\).

Step 1: Find the Common Ratio

To find the common ratio of the geometric series, divide one term by the previous term. Looking at the given series \(1 + x + x^2 + x^3 + \cdots\) it can be seen that each term is multiplied by \(x\) to get the next term. Therefore, the common ratio \(r\) is \(x\).

Step 2: Write the Function For the Sum

The sum \(S\) of an infinite geometric series is given by the formula \(S = a / (1 - r)\), where \(a\) is the first term and \(r\) is the common ratio. For the given series, \(a = 1\) and \(r = x\), so the sum is given by the function \(S = 1 / (1 - x)\).

Step 3: Graph the Function and Partial Sums

This step involves using a graphing utility to graph the sum function found in step 2, as well as the partial sums \(S_3\) and \(S_5\). The partial sum \(S_3 = 1 + x + x^2\), and \(S_5 = 1 + x + x^2 + x^3 + x^4\). By plotting these values on the graph, one can compare how the series converges to the sum as more terms are added.

Key Concepts

Common Ratio

In a geometric series, the common ratio plays a crucial role. It's the factor by which each term is multiplied to get to the next term in the series. To identify the common ratio ( r ) in any geometric series, simply divide one term by the term preceding it. For instance, in the series given as:

• 1,
• x,
• x^2,
• x^3,
• ...

Each term is derived by multiplying the previous term by x , making x the common ratio. Recognizing the common ratio is vital because it allows us to predict subsequent terms and analyze the behavior of the series. Once you know the ratio, you gain insights into how quickly the series grows or diminishes.

Partial Sums

A partial sum of a series is the sum of the first few terms of the series. In geometric series, partial sums offer a snapshot of how a series behaves as it progresses. Let's illustrate with partial sums \(S_3\) and \(S_5\), calculated from our series:

• S_3 = 1 + x + x^2
• S_5 = 1 + x + x^2 + x^3 + x^4

These sums show how the series builds up as you add more terms. By graphing \(S_3\) and \(S_5\), you can visualize how the series approaches the total sum. The closer the partial sum is to the actual sum of the series, the more accurate it becomes in representing the series as a whole. Watching these partial sums grow gives valuable clues on the convergence of the series.

Sum of an Infinite Series

Infinite series often present the intriguing question: Can an infinite amount of terms result in a finite sum? The answer is yes, under certain conditions. The sum of an infinite geometric series is determined by the formula: \(S = \frac{a}{1 - r}\). Where \(a\) is the first term and \(r\) is the common ratio. For a series like 1 + x + x^2 + x^3 + \cdots, where \(a = 1\) and \(r = x\), knowing that \(|x| < 1\) ensures convergence. The formula becomes: \[ S = \frac{1}{1-x} \] This result highlights key aspects of geometric series and serves as a practical tool for calculating their sum. Realizing the infinite span of terms can indeed sum up to a specific value is both fascinating and foundational in understanding series convergence.