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Magazine Chapter 12: Problem 59
Expand the partial sum and find its value. \(\sum_{n=1}^{5} \frac{n}{n+1}\)
Short Answer
Expert verified
The value of the sum is \(\frac{71}{20}\) or 3.55.
Step by step solution
01
Identify the Sum to Expand
The given sum is \(\sum_{n=1}^{5} \frac{n}{n+1}\). Start by identifying the sum from n = 1 to n = 5.
02
Write Individual Terms
Write the individual terms by substituting the values of n from the sum expression. These terms will be: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\).
03
Set Up the Expanded Sum
Set up the expanded sum by writing the terms together: \(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\).
04
Convert Fractions to Common Denominator
To add the fractions, they need to be converted to a common denominator. In this case, the least common multiple (LCM) of the denominators 2, 3, 4, 5, and 6 is 60.
05
Rewrite Each Fraction
Rewrite each fraction with the common denominator of 60: \(\frac{1}{2} = \frac{30}{60}\), \(\frac{2}{3} = \frac{40}{60}\), \(\frac{3}{4} = \frac{45}{60}\), \(\frac{4}{5} = \frac{48}{60}\), \(\frac{5}{6} = \frac{50}{60}\).
06
Add the Fractions
Add the numerators of the fractions with the common denominator: \(30 + 40 + 45 + 48 + 50 = 213\). So, the sum becomes \(\frac{213}{60}\).
07
Simplify the Fraction
Simplify the fraction \(\frac{213}{60}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3. This yields \(\frac{71}{20}\).
08
Convert to Decimal (Optional)
If needed, convert the fraction \(\frac{71}{20}\) to a decimal by dividing 71 by 20, which is approximately 3.55.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fraction Addition
Adding fractions may seem tricky, but with a few simple steps, you can master it! Each fraction has a numerator (top number) and a denominator (bottom number). To add fractions:
- First, ensure the denominators are the same.
- Next, add the numerators together.
- Finally, write the sum over the common denominator.
Common Denominator
Finding a common denominator is crucial for adding fractions. A common denominator is a shared multiple of the fractions' denominators.
- Identify the denominators of all fractions involved.
- Find the least common multiple (LCM) of these denominators.
Series Expansion
A series expansion involves expressing a sum of terms. Each term corresponds to a sequence in the series. For the partial sum \(\frac{n}{n+1}\), we substitute values of n to get individual terms: \[\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\]
To find the expanded sum, we write out all these terms and prepare to add them: \(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\). This process helps us break down and manage complex sums step by step.
To find the expanded sum, we write out all these terms and prepare to add them: \(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\). This process helps us break down and manage complex sums step by step.
Simplifying Fractions
Simplifying fractions makes them easier to work with. To simplify a fraction:
- Find the greatest common divisor (GCD) of the numerator and denominator.
- Divide both the numerator and the denominator by this GCD.
