Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, use the geometric series \(\sum_{k=1}^{\infty}\left(\frac{1}{2}\right)\). Graph the first 7 partial sums of the series.

Step-by-Step Solution

Answer

Plot the points (1, 0.5), (2, 1), (3, 1.5), (4, 2), (5, 2.5), (6, 3), (7, 3.5) to graph the partial sums.

Step 1: Understanding the Geometric Series

A geometric series is of the form \( a + ar + ar^2 + ar^3 + \, ... \), where \( a \) is the first term and \( r \) is the common ratio. For this series, \( a = \frac{1}{2} \) and \( r = 1 \).

Step 2: Finding Partial Sums

The partial sum of the first \( n \) terms of a geometric series is given by the formula: \( S_n = a \frac{1-r^n}{1-r} \). However, since \( r = 1 \) in this series, we simply keep adding \( a \) for each term, making the series additive rather than geometric since the ratio of commonality is not a proper geometric constant.

Step 3: Calculating the First 7 Partial Sums

Calculate each partial sum for each \( n \) from 1 to 7:- For \( n=1 \): \( S_1 = \frac{1}{2} \)- For \( n=2 \): \( S_2 = \frac{1}{2} + \frac{1}{2} = 1 \)- For \( n=3 \): \( S_3 = 1 + \frac{1}{2} = \frac{3}{2} \)- For \( n=4 \): \( S_4 = \frac{3}{2} + \frac{1}{2} = 2 \)- For \( n=5 \): \( S_5 = 2 + \frac{1}{2} = \frac{5}{2} \)- For \( n=6 \): \( S_6 = \frac{5}{2} + \frac{1}{2} = 3 \)- For \( n=7 \): \( S_7 = 3 + \frac{1}{2} = \frac{7}{2} \)

Step 4: Graphing the Partial Sums

Plot the partial sums on a graph with \( n \) (the number of terms) on the x-axis and \( S_n \) (the partial sum) on the y-axis. The points to plot are: (1, 0.5), (2, 1), (3, 1.5), (4, 2), (5, 2.5), (6, 3), (7, 3.5). Connect these points with line segments to show the progression of partial sums.

Key Concepts

Partial Sums

When analyzing a geometric series, it's essential to understand the concept of **partial sums**. The partial sum of a series represents the sum of the first \( n \) terms. In simpler terms, it's like pausing halfway through counting and asking, "How much have I summed up so far?"
Since the ratio \( r \) in our initial exercise is equal to 1, this series strays from traditional geometric series behavior, turning it into a simple repetitive addition. Each term simply adds the same amount, \( \frac{1}{2} \).
For example:

• The first term is \( \frac{1}{2} \), so the first partial sum \( S_1 = \frac{1}{2} \).
• Add another \( \frac{1}{2} \) to get the second partial sum: \( S_2 = 1 \).
• Keep adding \( \frac{1}{2} \) to find each subsequent partial sum.

So, calculating these partial sums involves adding the same number again and again. This aspect makes it distinct from a typical decreasing geometric series, where terms generally shrink in size, affecting the speed at which the series converges.

Graphing Series

Visualizing series with a graph is a helpful way to comprehend their behavior over time. Consider the partial sums of the series we discussed. We can plot these sums on a graph, providing a clear picture of how they incrementally grow. Here's a simple approach to graph such a progression:
On the x-axis, mark the number of terms, \( n \), while the y-axis will represent the partial sums, \( S_n \). For our particular series:

• At \( n = 1 \), plot (1, 0.5)
• At \( n = 2 \), plot (2, 1)
• At \( n = 3 \), plot (3, 1.5)
• At \( n = 4 \), plot (4, 2)
• Continue similarly until \( n = 7 \)

Once you've plotted these points, connect them with a line. This line reveals a linear trend, indicating a consistent addition of \( \frac{1}{2} \) at each step. Such linear growth is unique for this series due to its constant term repetition, unlike the exponential nature of typical geometric series.

Mathematical Series

A **mathematical series** is essentially the sum of terms from a sequence, serving as a fundamental concept in understanding mathematical progressions. In mathematics, you encounter various series, and the geometric series is one well-known type. Defined as \( a + ar + ar^2 + \ldots \), this sequence is characterized by its first term \( a \) and a constant multiplier called the common ratio, \( r \).
In geometric terms, every consecutive element is produced by multiplying the previous one by \( r \). However, the exercise provided poses an anomaly since \( r = 1 \). In this case, rather than diminishing or growing exponentially, the series becomes straightforward, repeatedly adding the same term.
While it deviates from the traditional geometric series pattern, this exercise illustrates the fluidity with which series behavior can be adapted based on parameters like \( r \). It also underscores how unifying and crucial series are across different branches of mathematics, offering insights into patterns, convergence, and the infinite.