Problem 76 - A Graphical Approach to College Algebra

Question

Use summation notation to write each series. Start the index at \(i=1\). $$-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\dots-\frac{1}{2187}$$

Step-by-Step Solution

Answer

\(\sum_{i=1}^{7} (-1)^{i+1} \frac{1}{3^i}\)

Step 1: Identify the Pattern

First, we need to identify the pattern in the series. The series is alternating with signs and appears to decrease in absolute value by a constant factor. It starts with \(-\frac{1}{3}\), then \(\frac{1}{9}\), and so on. We recognize that each term is of the form \((-1)^{i+1}\frac{1}{3^i}\).

Step 2: Determine the Last Term

Identify the last term of the series, which is \(-\frac{1}{2187}\). To find what power of 3 leads to 2187, we calculate: \(3^7 = 2187\). Thus, the last term corresponds to \(i=7\).

Step 3: Write the Series in Summation Notation

Now that we have the general term and the number of terms, we can write the series using summation notation: \[\sum_{i=1}^{7} (-1)^{i+1} \frac{1}{3^i}\]

Step 4: Verify the Summation

Verify that substituting \(i=1\) to \(i=7\) in the summation gives us the correct series terms. For example, when \(i=1\), the term is \(-\frac{1}{3}\), and when \(i=7\), the term is \(-\frac{1}{2187}\). This confirms our summation is correctly representing the series.

Key Concepts

Series

In mathematics, a series is the sum of the terms of a sequence. Sequences are ordered lists of numbers, and when you add up these numbers, you form what is called a series. A series can have a finite number of terms, called a finite series, or an infinite number of terms, called an infinite series. For example, the sequence \(1, 2, 3, \ldots, n\) can be summed up to form the series \(1 + 2 + 3 + \ldots + n\). Each series is defined by a rule or formula for its general term, often written using summation notation which employs the Greek capital letter sigma \(\Sigma\). This notation helps to efficiently express the sum without listing all terms, especially useful for long or infinite series.

The key concepts related to series include:

• **General term** \(a_n\): the formula giving the \(n^{th}\) term of the series.
• **Convergence**: whether an infinite series approaches a fixed sum.
• **Divergence**: when an infinite series does not settle into a fixed sum.

By understanding and calculating the common pattern within a sequence, we transform it into a series in summation notation, greatly simplifying mathematical operations and analysis.

Alternating Series

An alternating series is a particularly interesting type of series where the signs of the terms alternate between positive and negative. This creates a zigzag pattern of terms that differ fundamentally from series where all terms have the same sign. A simple alternating series has terms of the form \[(-1)^i a_i\]where \(a_i\) are the positive terms of the sequence.

They are important in mathematics due to their property of convergence under certain conditions, even when the series terms themselves do not head towards zero. An essential criterion for convergence of alternating series is known as the Alternating Series Test, which stipulates:

• The absolute value of the terms decreases monotonically (each term is smaller than the one before it).
• The terms approach zero as the index goes to infinity.

Using the Alternating Series Test, one can determine if these series converge to a limit, even if the exact sum is difficult to find. This functional characteristic is widely applied in approximations, such as Taylor and Fourier series, making alternating series a fundamental component of mathematical analysis.

Exponential Function

The exponential function is one of the most significant functions in mathematics, most commonly represented as \(e^x\) where \(e\) is approximately equal to 2.71828. This function models growth or decay processes, such as compound interest and population growth. It grows very quickly, multiplying by a fixed power, unlike polynomial or linear functions.

In series, exponential functions can also be represented as part of terms, especially when handling geometric series. In geometric sequences where each term is a constant multiple of the previous one, terms can sometimes contain powers, resembling exponential forms. For example, a geometric series with the general formula \(a_1 r^{n-1}\), where \(r\) is the common ratio, can rapidly grow or shrink based on \(r\).

Understanding the role of exponential functions within series allows for recognizing patterns and understanding growth models. In scientific computation, exponential functions are crucial due to their applicability in modeling real-world processes, from waves in physics to growth rates in biology. This function’s ability to model both small and very large changes, combined with their mathematical properties, makes them a centerpiece of both pure and applied mathematics.