Problem 7 - Thomas Calculus

Question

Which of the following express \(1+2+4+8+16+32\) in sigma notation? $$ a. \sum_{k=1}^{6} 2^{k-1} \quad \text { b. } \sum_{k=0}^{5} 2^{k} \quad \text { c. } \sum_{k=-1}^{4} 2^{k+1}$$

Step-by-Step Solution

Answer

All options a, b, and c are correct.

Step 1: Understand the Problem

We need to express the series \(1+2+4+8+16+32\) using sigma notation, identifying the correct expression among the given options.

Step 2: Identify the Pattern in the Series

Observe that each term in the series is a power of 2: \(1 = 2^0\), \(2 = 2^1\), \(4 = 2^2\), \(8 = 2^3\), \(16 = 2^4\), and \(32 = 2^5\). This confirms that the series follows the pattern \(2^k\) starting from \(k = 0\) to \(k = 5\).

Step 3: Evaluate Option a

Option a is \(\sum_{k=1}^{6} 2^{k-1}\). If we substitute \(k\) values from 1 to 6, we get \(2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}\), which matches the series \(1 + 2 + 4 + 8 + 16 + 32\). Hence, option a correctly represents the series.

Step 4: Evaluate Option b

Option b is \(\sum_{k=0}^{5} 2^{k}\). Substituting \(k\) values from 0 to 5 gives us \(2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}\). This also matches the series \(1 + 2 + 4 + 8 + 16 + 32\), so option b is another possible correct expression.

Step 5: Evaluate Option c

Option c is \(\sum_{k=-1}^{4} 2^{k+1}\). Substituting \(k\) values from -1 to 4 gives us \(2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}\). This again matches the series \(1 + 2 + 4 + 8 + 16 + 32\), so option c is also correct.

Step 6: Conclusion

All three options, a, b, and c, correctly express the series \(1 + 2 + 4 + 8 + 16 + 32\) in sigma notation. Each utilizes a different range or manipulation of the index \(k\) to represent the same list of terms.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This means that if you have a first term \(a\) and a common ratio \(r\), the series looks like \(a, ar, ar^2, ar^3, \ldots\). For the series \(1+2+4+8+16+32\), the first term is 1 and the common ratio is 2.
- First term: 1 - Common ratio: 2 - Sequence: \(1, 1\times2, 1\times2^2, 1\times2^3, \ldots\)
This makes it easy to calculate subsequent terms. Geometric series are useful in algebra and calculus, especially for finding the sum of infinitely long series, as the sum can be easily computed when the common ratio is between -1 and 1.

Mathematical Series

Mathematical series are expressions that sum numbers in a sequence. Series can take many forms and have different properties.

• Finite Series: A series with a limited number of terms. For example, \(1 + 2 + 4 + 8 + 16 + 32\) is a finite series with 6 terms.
• Infinite Series: A series with an unlimited number of terms, continuing indefinitely.
• Convergence: Some infinite series can converge, meaning they approach a certain value as more terms are added, like the geometric series \(a + ar + ar^2 + \ldots\).

Sigma notation is a common way to express series and can easily represent patterns using a simple and compact form. In our example, the series is expressed as \(\sum_{k=0}^{5} 2^{k}\), indicating that you sum up the terms starting from \(k = 0\) to \(k = 5\). This compact representation helps simplify complex expressions.

Power of Two

The 'power of two' refers to numbers that can be written as \(2^n\), where \(n\) is an integer. In binary systems and computer science, powers of two play a crucial role, as bits and bytes are based on this concept.
- **2⁰ = 1** - **2¹ = 2** - **2² = 4** - **2³ = 8** - **2⁴ = 16** - **2⁵ = 32**
In our series \(1+2+4+8+16+32\), each term is a power of two. Understanding this helps in identifying patterns quickly as it relies on a sequence of exponents. Powers of two increase exponentially, making them fundamental in doubling processes and growth calculations in various fields, from computer memory to geometry.