Problem 28 - Essential Calculus

Question

Numerical, Graphical, and Analytic Analysis (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series. $$\begin{array}{|l|l|l|l|l|l|}\hline n & 5 & 10 & 20 & 50 & 100 \\\\\hline S_{n} & & & & &\\\\\hline\end{array}$$ $$ \sum_{n=1}^{\infty} 3(0.85)^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is 20. The magnitude of the terms decreases as \(n\) increases so it takes a large number of terms for the partial sums to approach the total sum.

Step 1: Calculate the Sum of the Infinite Series

The given series is a geometric series with first term \(a = 3\) and common ratio \(r = 0.85\). The sum \(S\) of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\). Substituting the given values into the formula, we get \(S = \frac{3}{1 - 0.85} = 20\)

Step 2: Compute Indicated Partial Sums

For the partial sums, use the formula of the sum for a finite geometric series, \(S_n = \frac{a(1 - r^n)}{1 - r}\), to compute for \(n = 5, 10, 20, 50, 100\). Use a graphing utility to verify these computations.

Step 3: Graph the First 10 Terms of the Sequence and a Horizontal Line

Use a graphing utility to plot the first 10 terms of the sequence of partial sums. Then, draw a horizontal line at \(y = 20\) to represent the total sum.

Step 4: Explain the Relationship

Observe the graph and the magnitude of the terms of the series. Explain that as the terms get smaller in magnitude (due to \(0.85^n\)), the rate at which the partial sums approach the total sum slows down. Despite this, because the series is convergent, the partial sums will eventually reach the total sum, as evidenced by the horizontal line on the graph.

Key Concepts

Infinite Series

An infinite series is what you get when you sum up infinitely many terms of a sequence. In this case, we're dealing with a special kind of infinite series known as a geometric series, where every term is a constant multiple of the previous one. For a geometric series to have a sum, the common ratio, denoted as \( r \), must be between -1 and 1. This ensures that the terms get smaller and smaller, eventually adding up to a finite total. In our exercise, the given series is \( \sum_{n=1}^{\infty} 3(0.85)^{n-1} \). To find the sum of such an infinite geometric series, we use the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term, and \( r \) is the common ratio. Substituting the known values - \( a = 3 \) and \( r = 0.85 \) - into this formula gives us \( S = \frac{3}{1 - 0.85} = 20 \). This means if you were to add an infinite number of terms from this series, their sum would approach 20.

Partial Sums

Partial sums are the sums of the first \( n \) terms of a series. Think of them as checkpoints to see how close you are getting to the actual sum of the infinite series. In our exercise, partial sums are calculated using the formula for the sum of a finite geometric series: \( S_n = \frac{a(1 - r^n)}{1 - r} \). You compute this for different values of \( n \), such as 5, 10, 20, 50, and 100, which gives insight into how the series behaves as more terms are added. As \( n \) increases, \( r^n \) becomes very small, causing the partial sums \( S_n \) to inch closer and closer to the sum of the infinite series. Therefore, understanding partial sums helps us see how the series reaches or approaches its infinite sum over time.

Sequence Convergence

Sequence convergence is all about determining whether the series approaches a particular value as more terms are added. For a geometric series like ours, if the common ratio \( r \) is between -1 and 1, the series is convergent. This simply means the sum approaches a finite number as you continue to add terms. In our exercise, the common ratio \( r = 0.85 \) satisfies this condition, indicating that the series converges to 20.

• As terms get progressively smaller due to multiplying by \( 0.85 \), each subsequent term has less impact on the total sum.
• The partial sums approach but never exceed the infinite sum, illustrating convergence.

Understanding convergence gives confidence that adding more and more terms isn't creating instability or infinite sums, but rather it's smoothly closing in on a specific value.

Graphical Analysis

Graphical analysis helps visualize how the partial sums behave as more terms are added. By plotting the first 10 partial sums on a graph, you see them slowly climbing towards the value of 20, which is the sum of the infinite series. You also add a horizontal line at \( y = 20 \) that serves as a visual guide for the ultimate sum of the series.

• Initially, the partial sums rise steeply this shows a fast accumulation of sums in the early terms.
• As more terms get added, the curve flattens, reflecting the slowing contribution of each new term.
• Eventually, the partial sums closely hover around the horizontal line, showing they are nearly reaching or have reached the sum.

This graphical approach provides a clear representation of the progression of partial sums towards convergence, as opposed to only calculating and comparing numbers.