Problem 106 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}.\)

Step-by-Step Solution

Answer

The statement is false. The correct statement should be, 'The sum of the geometric series can be determined without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\).' The sum of the series is \(\frac{1023}{1024}.\)

Step 1: Identify the first term, common ratio and the number of terms

The first term \(a\) is \(\frac{1}{2}\). The ratio can be figured out by taking the second term \(\frac{1}{4}\) and dividing it by the first term \(\frac{1}{2}\), which results in \(r = \frac{1}{2}\). The final term in the series is \(\frac{1}{512}\). To find the number of terms (\(n\)), take the log base \(\frac{1}{2}\) of both the first term and the last term and add 1 to the result. Hence \(n = \log_{\frac{1}{2}}{\frac{1}{2}} - \log_{\frac{1}{2}}{\frac{1}{512}} + 1 = 10.\

Step 2: Substituting the values into the formula

Substitute the values of \(a = \frac{1}{2}, r = \frac{1}{2}, n = 10\) into the formula for sum of geometric series. Doing this yields \( S = \frac{1}{2} \cdot \frac{1 - (\frac{1}{2})^{10}}{1 - \frac{1}{2}}\)

Step 3: Calculating the sum

Perform the calculation to evaluate the sum. This results in \(S = 1 - \frac{1}{1024} = \frac{1023}{1024}\)

Step 4: Evaluating the statement

Because we can calculate the exact sum of the geometric series without knowing the intermediate terms, the original statement is false. It will be made true by removing the word 'estimated'. So the correct statement would be: 'The sum of the geometric series can be determined without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}.\)'

Key Concepts

Geometric Progression

Understanding geometric progressions is pivotal in grasping a range of mathematical concepts. A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. These progressions have diverse applications, from computing interest in finance to modeling natural phenomena in science.

For a geometric series, where individual terms of a geometric progression are added together, we can express it as:
\[ S = a_1 + a_1r + a_1r^2 + ... + a_1r^{n-1} \]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. One fascinating property of geometric series is that if \(|r| < 1\), the series has a finite sum, even as \(n\) approaches infinity. This introduces an important aspect: the convergence of geometric series.

Series Convergence

Knowing when a series converges is essential for understanding its behavior and being able to calculate its sum precisely. Convergence in the context of series refers to the idea that as we add more and more terms, the series approaches a specific value. In contrast, a series that doesn't converge is said to diverge, which means it doesn't settle towards any finite value.

For a geometric series with a common ratio \(|r| < 1\), the sum converges to a finite value given by the formula:
\[ S = \frac{a_1}{1 - r} \]
This is only valid if the series is infinite. However, if we're dealing with a finite number of terms, as we often do in practice, the sum of the first \(n\) terms from a geometric series is found using this formula:
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
Applying this formula requires knowledge of the first term, the common ratio, and the number of terms. The convergence of a series greatly impacts various fields such as engineering and physics, where the concept is necessary to predict long-term behavior from short-term patterns.

Exponents and Logarithms

Exponents and logarithms are like two sides of the same coin in mathematics. They are interconnected and vital tools for solving equations involving exponential growth or decay. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\).

Logarithms, on the other hand, answer the question: to what exponent must we raise a given base to produce a certain number? For instance, if we have \(2^3 = 8\), the logarithm form would be \(log_2 8 = 3\), meaning we raise 2 to the exponent of 3 to get 8.

Understanding logarithms is critical when determining the number of terms in a geometric progression. In the above exercise, we used the logarithmic property to find the number of terms in the series by solving \(log_{\frac{1}{2}}{\frac{1}{512}}\). In essence, exponents and logarithms help us navigate through problems involving geometric progressions and series more effectively. This connection is not only fascinating but also showcases the beautiful symmetry inherent in mathematics.