Problem 20 - Calculus

Question

Evaluate each sum. (a) \(\sum_{m=2}^{4}\left(\frac{1}{m}\right)\) (b) \(\sum_{i=1}^{6}(2-i)\) (c) \(\sum_{k=0}^{4} \cos \left(\frac{k \pi}{4}\right)\)

Step-by-Step Solution

Answer

(a) 1.083; (b) -9; (c) 0.

Step 1: Understanding the Summation Process

To solve the given problems, we will evaluate each sum by substituting the values of the index variable within the given limits and then finding the sum of the series.

Step 2: Calculating Sum (a)

Evaluate \( \sum_{m=2}^{4} \left( \frac{1}{m} \right) \). Substitute \( m = 2, 3, \) and \( 4 \) into the expression: \( \frac{1}{2} + \frac{1}{3} + \frac{1}{4}. \) Calculate each fraction and sum them.\[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = 0.5 + 0.333 + 0.25 = 1.083 \]

Step 3: Calculating Sum (b)

Evaluate \( \sum_{i=1}^{6} (2-i) \). Substitute \( i = 1, 2, 3, 4, 5, \) and \( 6 \) into the expression: \( (2-1) + (2-2) + (2-3) + (2-4) + (2-5) + (2-6) \). Perform the calculations:\[ 1 + 0 - 1 - 2 - 3 - 4 = -9 \]

Step 4: Calculating Sum (c)

Evaluate \( \sum_{k=0}^{4} \cos \left( \frac{k \pi}{4} \right) \). Substitute \( k = 0, 1, 2, 3, \) and \( 4 \) into the expression: \( \cos(0), \cos\left(\frac{\pi}{4}\right), \cos\left(\frac{\pi}{2}\right), \cos\left(\frac{3\pi}{4}\right), \cos(\pi) \).These values are: \( 1 + \frac{\sqrt{2}}{2} + 0 - \frac{\sqrt{2}}{2} - 1 \). Perform the calculations:\[ 1 + 0.707 - 0 - 0.707 - 1 = 0 \]

Step 5: Concluding the Results

Therefore, after evaluating each individual sum, we have the following results: - (a) is approximately 1.083 - (b) is -9 - (c) is 0

Key Concepts

Series Evaluation

Series evaluation is the process of calculating the sum of a sequence of terms defined by a general rule or formula. In mathematics, a series is the sum of the elements of a sequence. Understanding how to evaluate a series is crucial for solving various mathematical problems. The process starts by identifying the limits of the summation and the formula for the terms involved. For example, in \(\sum_{m=2}^{4}\left(\frac{1}{m}\right)\), you consider values from \((m)\) starting at 2 to 4. For each step:

• Substitute the index value into the formula.
• Calculate each individual term.
• Add them up to get the final result of the series.

Using the above series, substituting the values gives us fractions \(\frac{1}{2}, \frac{1}{3}, \) and \(\frac{1}{4}\). Summing them results in approximately 1.083. Remember to always check your calculations for accuracy.

Trigonometric Summation

Trigonometric summation involves sums where each term of the series is derived using trigonometric functions, such as sine or cosine. These functions are periodic and can result in interesting series properties due to their cyclical nature. In the problem \(\sum_{k=0}^{4} \cos \left(\frac{k \pi}{4}\right)\), substituting values for \(k\) involves utilizing trigonometric identities:

• \(\cos(0) = 1\)
• \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
• \(\cos\left(\frac{\pi}{2}\right) = 0\)
• \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)
• \(\cos(\pi) = -1\)

Adding these values gives you a sum of 0, showing the self-canceling nature of trigonometric series due to the inherent symmetry of cosine function values.

Index Substitution

Index substitution is a crucial step in calculating series, as it involves replacing the index variable within a defined range into the expression. This is essential for determining the actual terms of the series to be summed. Consider the series \(\sum_{i=1}^{6}(2-i)\). Here’s how index substitution works:

• Identify the range for the index, in this case, 1 through 6.
• Substitute each index value back into the formula, \(2-i\).
• Calculate resulting values: \(1, 0, -1, -2, -3, -4\).
• Add these results sequentially to find the total sum, which results in -9.

By systematically substituting the index and evaluating, you can tackle any given series regardless of its complexity. It’s important to carefully follow the order and ensure no substitutions are skipped to maintain accuracy.