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Magazine Chapter 12: Problem 44
Find the sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{512} $$
Short Answer
Expert verified
The sum of the series is \( \frac{341}{512} \).
Step by step solution
01
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 \( n \geq 1 \).
02
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 \( \frac{1}{2^0} \).
03
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 \( -\frac{1}{2^9} \).
04
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 \( n = 10 \).
05
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:\[S_{10} = \frac{1(1 - \frac{1}{1024})}{1 + \frac{1}{2}} = \frac{1 \times \frac{1023}{1024}}{\frac{3}{2}} = \frac{1023}{1536}.\]
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
