Problem 44 - Algebra and Trigonometry

Question

Find the sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{512} $$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{341}{512} \).

Step 1: Identify the Sequence

The given series is an alternating series: \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots - \frac{1}{512} \). Each term after the first is of the form \( (-1)^{n}\frac{1}{2^{n}} \), where \( n \geq 1 \).

Step 2: Find the General Term

The general term of the series is \( a_n = (-1)^{n} \cdot \frac{1}{2^{n}} \), where \( n \geq 0 \) and the first term is 1, which is \( \frac{1}{2^0} \).

Step 3: Determine the Number of Terms

Observe the last term provided \( -\frac{1}{512} \), which can be written as \( -\frac{1}{2^9} \). This indicates there are 10 terms, starting from \( 1 = \frac{1}{2^0} \) to \( -\frac{1}{2^9} \).

Step 4: Apply the Sum Formula for a Finite Geometric Series

This is a geometric series where the common ratio \( r = -\frac{1}{2} \) and the number of terms, \( n = 10 \). The sum of a geometric series is given by:\[S_n = \frac{a(1 - r^n)}{1 - r},\]where \( a = 1 \), \( r = -\frac{1}{2} \) and \( n = 10 \).

Step 5: Calculate the Sum

Plug in the given values into the formula:\[S_{10} = \frac{1(1 - (-\frac{1}{2})^{10})}{1 - (-\frac{1}{2})}.\]Calculate \((-\frac{1}{2})^{10} = \frac{1}{1024}\), and:\[S_{10} = \frac{1(1 - \frac{1}{1024})}{1 + \frac{1}{2}} = \frac{1 \times \frac{1023}{1024}}{\frac{3}{2}} = \frac{1023}{1536}.\]

Step 6: Simplify the Sum

Simplify \( \frac{1023}{1536} \):Both 1023 and 1536 can be divided by their greatest common divisor (GCD), which is 3. \[\frac{1023}{1536} = \frac{1023 \div 3}{1536 \div 3} = \frac{341}{512}.\]This fraction is already in its simplest form.

Key Concepts

Alternating Series

In mathematics, an alternating series is a series where the signs of each term alternates between positive and negative. In the given problem, we have an alternating series: \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots - \frac{1}{512} \). Here's how it works:

• The alternation happens because each term's sign is determined by \((-1)^n\).
• If \(n\) is even, the term is positive. Conversely, if \(n\) is odd, the term is negative.

This pattern is critical in identifying the behavior and structure of the series, offering insight into its eventual sum. Alternating series can converge even if individual terms do not approach zero as they alternate, helping balance out the sums.

Sequence Identification

Determining the characteristics of a sequence is key to solving series problems. Here, each term fits the formula: \( a_n = (-1)^n \cdot \frac{1}{2^n} \).

• The factor \( (-1)^n \) decides the term's sign, as explained in the alternating series.
• The fraction \( \frac{1}{2^n} \) reflects how each term's magnitude is calculated, systematically decreasing as \(n\) increases.

Being able to establish this correlation is crucial for any further calculation or application of formulas related to series, such as summation. Recognizing the sequential order and mathematical pattern aids in efficiently applying relevant formulae and solving the problem.

Sum of a Series

The essence of finding the sum of a series lies in capturing its complete value through a summation formula. In this exercise, a finite geometric series formula is applied to determine the total sum.The sum of a finite series can be calculated using:\[ S_n = \frac{a(1 - r^n)}{1 - r} \]

• \(a\) is the first term, which is 1.
• \(r\) is the common ratio, here \( -\frac{1}{2} \).
• \(n\) is the total number of terms, which we found to be 10.

Applying these values yields the series' sum efficiently. Practically, this approach simplifies complex series into manageable calculations, demonstrating the value of using series formulas.

Finite Geometric Series

Finite geometric series involve the summation of a finite number of terms, each derived by multiplying a consistent factor, known as the common ratio.For this particular series:

• It starts with 1, denoted as \( a \) in the formula.
• The common ratio, \( r \), is \( -\frac{1}{2} \).
• The series ends after 10 terms.

The uniqueness of a finite geometric series is that the number of terms is limited, and the rate of change is constant. This type of series can be comprehensively analyzed and computed using the formula:\[ S_n = \frac{a(1 - r^n)}{1 - r} \].Understanding these characteristics is essential for evaluating and solving problems involving geometric progressions.