Let [tex]$b_n$[/tex] be an infinite sequence of zeros and ones. What is the largest possible value of [tex]$x=\sum_{n=1}^{\infty} \frac{b_n}{2^n}$[/tex]?
Answer (2)
To maximize the sum, set each b n to 1.
Recognize the sum as a geometric series: ∑ n = 1 ∞ 2 n 1 .
Apply the formula for the sum of an infinite geometric series: S = 1 − r a .
Calculate the sum: S = 1 − 2 1 2 1 = 1 . The largest possible value of x is 1 .
Explanation
Understanding the Problem We are given an infinite sequence b n of zeros and ones, and we want to find the largest possible value of the sum x = ∑ n = 1 ∞ 2 n b n . This means each b n can be either 0 or 1. Our goal is to maximize the value of x .
Maximizing the Sum To maximize the sum, we should choose the largest possible value for each b n . Since b n can only be 0 or 1, we should set b n = 1 for all n . This gives us the series: x = n = 1 ∑ ∞ 2 n 1 = 2 1 + 4 1 + 8 1 + 16 1 + ⋯
Identifying Geometric Series The series is a geometric series with the first term a = 2 1 and the common ratio r = 2 1 . The sum of an infinite geometric series is given by the formula: S = 1 − r a
Calculating the Sum Plugging in the values for a and r , we get: S = 1 − 2 1 2 1 = 2 1 2 1 = 1
Final Answer Therefore, the largest possible value of x is 1.
Examples
Imagine you're designing a system where each bit you transmit has a decreasing value (half of the previous bit). This problem shows that even with an infinite number of bits, if each bit is either 'on' (1) or 'off' (0), the maximum value you can represent is 1. This concept is useful in digital communication, data compression, and computer science where understanding the limits of infinite series is crucial.
The largest possible value of the sum x = ∑ n = 1 ∞ 2 n b n is achieved when each b n is set to 1. This leads to the sum of an infinite geometric series, which calculates to 1 . In summary, the maximum value of x is 1 .
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