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Chapter 14: Problem 69

Write out and evaluate each sum. $$ \sum_{k=3}^{5} \frac{(-1)^{k}}{k(k+1)} $$

Short Answer

Expert verified
\( \frac{-1}{15} \)

Step by step solution

01

- Identify the limits and function

The given sum is from 3 to 5 for the function \ \ \( \frac{(-1)^{k}}{k(k+1)} \). This means we need to evaluate the sum for \( k = 3, 4, 5 \).
02

- Evaluate the function for each value of k

For \( k = 3 \), the value is \( \frac{(-1)^{3}}{3(3+1)} = \frac{-1}{3 \cdot 4} = \frac{-1}{12} \). For \( k = 4 \), the value is \( \frac{(-1)^{4}}{4(4+1)} = \frac{1}{4 \cdot 5} = \frac{1}{20} \). For \( k = 5 \), the value is \( \frac{(-1)^{5}}{5(5+1)} = \frac{-1}{5 \cdot 6} = \frac{-1}{30} \).
03

- Sum the evaluated terms

Sum the evaluated terms: \( \frac{-1}{12} + \frac{1}{20} + \frac{-1}{30} \). First, convert them to a common denominator. The least common multiple of 12, 20, and 30 is 60. \( \frac{-1}{12} = \frac{-5}{60} \), \( \frac{1}{20} = \frac{3}{60} \), and \( \frac{-1}{30} = \frac{-2}{60} \). So, summing them: \( \frac{-5}{60} + \frac{3}{60} + \frac{-2}{60} = \frac{-5 + 3 - 2}{60} = \frac{-4}{60} = \frac{-1}{15} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

limits of summation
In summation notation, the limits of summation define the range of values for the variable involved in the summation. For our example sum oindent\textbf{ \( \text{ \text{ } } \frac{(-1)^{k}}{k(k+1)} \text{ from values k = 3 to 5 \) } }, the limits are 3 and 5. This tells us that we need to evaluate the given function for each integer value of \(k\) starting from 3 and ending at 5.
evaluating function
Evaluating a function within a sum means computing the function's value for each specific instance of the variable within the given limits. In our case: \( k = 3, 4, 5 \). For each of these values of \( k \), we plug them into the function \( \frac{(-1)^k}{k(k+1)} \) and compute the result:
  • For \( k = 3 \): \( \frac{(-1)^3}{3(3+1)} = \frac{-1}{12} \)
  • For \( k = 4 \): \( \frac{(-1)^4}{4(4+1)} = \frac{1}{20} \)
  • For \( k = 5 \): \( \frac{(-1)^5}{5(5+1)} = \frac{-1}{30} \)
common denominators
When summing fractions, it's important to have a common denominator. This makes adding the fractions straightforward. In our example, \( \frac{-1}{12} + \frac{1}{20} + \frac{-1}{30} \) We need to convert these fractions to have the same denominator before we can sum them. The least common multiple of 12, 20, and 30 is 60.
  • Convert each fraction: \( \frac{-1}{12} = \frac{-5}{60} \) \( \frac{1}{20} = \frac{3}{60} \) \( \frac{-1}{30} = \frac{-2}{60} \)
    • Then we sum them up:\( \frac{-5}{60} + \frac{3}{60} + \frac{-2}{60} \) Combine the numerators: \( \frac{-5 + 3 - 2}{60} = \frac{-4}{60} \) Simplify to get the final result: \(\frac{-1}{15} \).

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