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Magazine Chapter 14: Problem 64
Write out and evaluate each sum. $$ \sum_{k=2}^{5} \frac{k-1}{k+1} $$
Short Answer
Expert verified
The sum evaluates to 2.1.
Step by step solution
01
Understand the Sum Notation
Understand that the given sum \(\sum_{k=2}^{5} \frac{k-1}{k+1} \) means you need to evaluate the expression \(\frac{k-1}{k+1}\) for each integer value of \(k\) from 2 to 5 and then add all these values together.
02
Calculate the Terms Individually
Calculate the value of \(\frac{k-1}{k+1}\) for each \(k\) from 2 to 5: \[\begin{align*}\text{For } k=2, \quad \frac{2-1}{2+1} &= \frac{1}{3}, \ \text{For } k=3, \quad \frac{3-1}{3+1} &= \frac{2}{4} = \frac{1}{2}, \ \text{For } k=4, \quad \frac{4-1}{4+1} &= \frac{3}{5}, \ \text{For } k=5, \quad \frac{5-1}{5+1} &= \frac{4}{6} = \frac{2}{3}.\end{align*}\]
03
Add the Calculated Terms
Add the individual terms obtained: \[\frac{1}{3} + \frac{1}{2} + \frac{3}{5} + \frac{2}{3}.\]
04
Find a Common Denominator
Convert each fraction to have a common denominator, which is 30: \[\begin{align*}\frac{1}{3} &= \frac{10}{30}, \ \frac{1}{2} &= \frac{15}{30}, \ \frac{3}{5} &= \frac{18}{30}, \ \frac{2}{3} &= \frac{20}{30}. \end{align*}\]
05
Sum the Equivalent Fractions
Add the fractions with a common denominator: \[\begin{align*}\frac{10}{30} + \frac{15}{30} + \frac{18}{30} + \frac{20}{30} &= \frac{10 + 15 + 18 + 20}{30} \ &= \frac{63}{30} \ &= \frac{21}{10} \ &= 2.1.\end{align*}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Fractions
Fractions represent parts of a whole. They consist of a numerator (top part) and a denominator (bottom part). For example, in the fraction \(\frac{3}{5}\), 3 is the numerator and 5 is the denominator. Fractions can often be simplified by finding common factors of the numerator and the denominator. When adding or subtracting fractions, as you'll see, it's essential to have a common denominator. This ensures that each fraction represents parts of the same whole.
Finding Common Denominators
To add or subtract fractions, you need a common denominator. The common denominator is a shared multiple of the fractions' denominators. This ensures the fractions are comparable, representing the same sized parts of the whole.
For example, if you have \(\frac{1}{3}\) and \(\frac{1}{2}\), you cannot directly add them because the parts are different sizes.
The lowest common multiple (LCM) of 3 and 2 is 6. To convert these fractions: \(\frac{1}{3} = \frac{2}{6}\) and \(\frac{1}{2} = \frac{3}{6}\). Now, you can add them:
For example, if you have \(\frac{1}{3}\) and \(\frac{1}{2}\), you cannot directly add them because the parts are different sizes.
The lowest common multiple (LCM) of 3 and 2 is 6. To convert these fractions: \(\frac{1}{3} = \frac{2}{6}\) and \(\frac{1}{2} = \frac{3}{6}\). Now, you can add them:
- \(\frac{2}{6} + \frac{3}{6} = \frac{5}{6}\)
Evaluating Series
When you see summation notation like \(\begin{aligned}\sum_{k=2}^{5} \frac{k-1}{k+1}\text,&\end{aligned}\), it means to calculate the expression \(\frac{k-1}{k+1}\) for each value of k from 2 to 5, then sum all those results up.
For this example:
For this example:
- For\(\ k = 2\), \(\frac{2-1}{2+1} = \frac{1}{3}\)
- For\(\ k = 3\), \(\frac{3-1}{3+1} = \frac{2}{4} = \frac{1}{2}\)
- For\(\ k = 4\), \(\frac{4-1}{4+1} = \frac{3}{5}\)
- For\(\ k = 5\), \(\frac{5-1}{5+1} = \frac{4}{6} = \frac{2}{3}\)
Working with Arithmetical Sums
After calculating individual terms in a series, the next step often involves summing them. For the series \(\frac{1}{3} + \frac{1}{2} + \frac{3}{5} + \frac{2}{3}\), you need to add these fractions.
First, find a common denominator \(e.g., 30 in this case\), and convert each fraction:
Hence, the sum of the series is 2.1.
First, find a common denominator \(e.g., 30 in this case\), and convert each fraction:
- \(\frac{1}{3} = \frac{10}{30}\)
- \(\frac{1}{2} = \frac{15}{30}\)
- \(\frac{3}{5} = \frac{18}{30}\)
- \(\frac{2}{3} = \frac{20}{30}\)
Hence, the sum of the series is 2.1.
