Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Sequence of Partial Sums Use a graphing utility to create a table showing the sequence of the first 10 partial sums \(S_{1}, S_{2}, S_{3}, \ldots\) and \(S_{10}\) for the series. $$\sum_{n=1}^{\infty} 4(0.2)^{n-1}$$

Step-by-Step Solution

Answer

The sequence of the first 10 partial sums for the series \(\sum_{n=1}^{\infty} 4(0.2)^{n-1}\) is \(S_{1} = 4\), \(S_{2} = 4.8\), \(S_{3} = 4.96\), \(S_{4} = 4.992\), \(S_{5} = 4.9984\), \(S_{6} = 4.99968\), \(S_{7} = 4.999936\), \(S_{8} = 4.9999872\), \(S_{9} = 4.99999744\), and \(S_{10} = 4.999999488\).

Step 1: Identify parameters of the geometric series

The first term \(a\) of the series is 4 and the common ratio \(r\) is 0.2.

Step 2: Calculate each partial sum

Use the formula for the partial sum of a geometric series \(\(S_n = a(1 - r^n) / (1 - r)\) to compute each partial sum from \(S_1\) to \(S_{10}\). For example, to find \(S_1\), plug \(n = 1\) into the formula to get \(S_1 = 4\). To find \(S_2\), plug \(n = 2\) to get \(S_2 = 4 + 4*0.2 = 4.8\), and so on till \(S_{10}\).

Step 3: List the sequence of partial sums

After calculating each sum from \(S_1\) to \(S_{10}\), list them in a sequence to represent the sequence of partial sums.

Key Concepts

Understanding Partial Sums

Partial sums in a series are crucial to understanding how a sequence approaches its total sum. In a geometric series, a partial sum, denoted as \(S_n\), represents the sum of the first \(n\) terms. It provides an insight into the behavior of the series as you sum more terms.

For example, when starting with the first term \(S_1\), you only include the very first expression of the series. With each subsequent partial sum, you add one more term from the series. This accumulation builds a sequence of sums like \(S_1, S_2, S_3,\) and so on, leading up to \(S_{10}\) in this exercise.

• \(S_1 = 4\)
• \(S_2 = 4 + 0.8\)
• \(S_3 = 4 + 0.8 + 0.16\)

This shows how each subsequent sum builds upon the previous one, moving closer to the overall convergence of a potentially infinite series.

Sequence in the Context of Series

A sequence is an ordered list of numbers following a particular pattern. In the context of a geometric series, each term is derived by multiplying the preceding term with a constant called the common ratio. Sequences are fundamental in forming series like geometric series, where each successive term maintains a fixed ratio with the prior term.

The sequence in this context begins by defining the first term as \(a = 4\). As each term is multiplied by the common ratio \(r = 0.2\), it unfolds a series of numbers:

• First term: 4
• Second term: 0.8 (or \(4 \times 0.2\))
• Third term: 0.16 (or \(0.8 \times 0.2\))

By outlining the sequence from \(S_1\) to \(S_{10}\), students can observe how the sequence grows, portraying the decreasing trend typical in geometric sequences with common ratios less than 1. The sequence gets smaller with each term and illustrates how it contributes to forming the total sum of the series.

Examining the Common Ratio

The common ratio \(r\) in a geometric series is a constant factor between each term. In our exercise, the common ratio is 0.2, which plays a crucial role in determining the behavior and convergence of the series.

On applying this ratio:

• We see that each term of the series is 0.2 times the preceding term.
• This multiplier ensures the sequence formed by the series decreases exponentially as it progresses.
• When \(r\) is smaller than 1, as in this exercise, the series converges, and the terms progressively approach zero.

Recognizing the value of the common ratio aids in predicting whether a series will converge or diverge and helps visualize how quickly terms decrease. The smaller the common ratio, the faster the terms decline toward zero, leading to a quicker convergence towards a finite sum.

Utilizing a Graphing Utility

Graphing utilities can be invaluable tools in visualizing sequences and series. By tampering with these tools, students can understand partial sums more concretely. It aids them in seeing how the sum evolves with each added term.

To use a graphing utility:

• Begin by inputting the formula for the partial sum \(S_n = a(1 - r^n) / (1 - r)\).
• Create a table within the graphing utility to calculate each partial sum, from \(S_1\) to \(S_{10}\).
• Plot these sums to visualize how they approach the series' limit.

This visual representation helps grasp complex mathematical concepts more intuitively. It provides an interactive method for students to truly understand how the geometric series behaves, offering a stepping stone from abstract numbers to visible trends.