Problem 7 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Options a and b correctly express the series in sigma notation.

Step 1: Understand the Problem

We need to express the series \(1+2+4+8+16+32\) using sigma notation. This is a geometric series where each term is a power of 2.

Step 2: Identify the Pattern

The given series \(1+2+4+8+16+32\) corresponds to the sequence \(2^0, 2^1, 2^2, 2^3, 2^4, 2^5\). Here, \(2^0 = 1, 2^1 = 2,\) etc.

Step 3: Translate Series to Sigma Notation

The sigma notation for the sequence can be written as \(\sum_{k=a}^{b} 2^{k}\), where \(a\) is the starting exponent and \(b\) is the ending exponent.

Step 4: Analyze Given Options

Let's evaluate each option:- Option a: \(\sum_{k=1}^{6} 2^{k-1}\). For \(k=1\), it represents \(2^{1-1} = 2^0 = 1\), and for \(k=6\), it represents \(2^{6-1} = 2^5 = 32\). This matches the series.- Option b: \(\sum_{k=0}^{5} 2^{k}\). For \(k=0\), it represents \(2^{0} = 1\), and for \(k=5\), it represents \(2^{5} = 32\). This also matches the series.- Option c: \(\sum_{k=-1}^{4} 2^{k+1}\). For \(k=-1\), it represents \(2^{-1+1} = 2^0 = 1\), and for \(k=4\), it represents \(2^{4+1} = 2^5 = 32\), but series is incorrect as the series should be ending with \(k=6\).

Step 5: Choose the Correct Option

Options a and b both correctly express the series in sigma notation as each evaluates to the series \(1+2+4+8+16+32\). However, given that both match, we can choose either as a correct expression. Option c is incorrect.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant. This constant is called the common ratio. In our example, the series is \(1 + 2 + 4 + 8 + 16 + 32\). Each number in this series is obtained by multiplying the previous number by 2. Thus:

• The first term is 1 (\(2^0\))
• The second term is 2 (\(2^1\))
• The third term is 4 (\(2^2\))
• The fourth term is 8 (\(2^3\))
• The fifth term is 16 (\(2^4\))
• The sixth term is 32 (\(2^5\))

Geometric series are important in mathematics because they have a clear pattern and are easily represented in sigma notation.

Power of Two

The term 'Power of Two' refers to integers that can be written as some power of the number 2. Such numbers are frequently encountered in computer science, as they align with binary operations. For example, in our series, each number is a power of two:

• \(2^0 = 1\)
• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)

Understanding powers of two helps in simplifying and working with geometric series efficiently. It also aids in translating sequences to sigma notation by easily identifying the exponents.

Sequence Translation

Sequence translation involves expressing a series in a different mathematical form for easier manipulation or understanding. In this exercise, we need to translate a basic series into sigma notation.Sigma notation is a way of writing long sums in a compact and understandable form. The given series \(1 + 2 + 4 + 8 + 16 + 32\) is expressed in terms of \(2^k\) in sigma notation. Here are some examples to illustrate this:

• \(\sum\_{k=0}^{5} 2^k\)
• \(\sum\_{k=1}^{6} 2^{k-1}\)

Each style of sigma notation organizes and condenses the series, making it easier to identify properties and the number of terms involved.

Series Evaluation

Series evaluation involves summing up all the terms of a series to find a total. When using sigma notation, it becomes more straightforward to compute large sums without listing every individual term.To evaluate a geometric series quickly, you can use formulas or recognize patterns. For a geometric series like \(1 + 2 + 4 + 8 + 16 + 32\), there are practical approaches:

• Identify it as a sum of powers of two.
• Realize that each subsequent term is double the previous one.
• Utilize a formula: the sum \(S_n = a(1 - r^n)/(1 - r)\) where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the total number of terms.

By understanding sigma notation and geometric series, you make series evaluation manageable and efficient.