Evaluate the Summation sum from k=1 to 4 of 1/(3k)
The question asks you to calculate the total sum of a series where the terms are in the form of 1/(3k). You are to find the sum of these terms starting from where k is equal to 1 and ending when k is equal to 4. Each term in the summation is the reciprocal of three times the value of k.
$\sum_{k = 1}^{4} \frac{1}{3 k}$
Answer
Solution:
Step 1: Write out the series for each value of \( k \).
\[ \frac{1}{3 \cdot 1} + \frac{1}{3 \cdot 2} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4} \]
Step 2: Begin simplification.
Step 2.1: Break down each term individually.
Step 2.1.1: Remove the common factor of 1 where applicable.
\[ \frac{\cancel{1}}{3 \cdot \cancel{1}} + \frac{1}{3 \cdot 2} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4} \]
Step 2.1.1.1: Rewrite the simplified terms.
\[ \frac{1}{3} + \frac{1}{3 \cdot 2} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4} \]
Step 2.1.2: Calculate \( 3 \times 2 \).
\[ \frac{1}{3} + \frac{1}{6} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4} \]
Step 2.1.3: Calculate \( 3 \times 3 \).
\[ \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{3 \cdot 4} \]
Step 2.1.4: Calculate \( 3 \times 4 \).
\[ \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} \]
Step 2.2: Work towards a common denominator.
Step 2.2.1: Scale \( \frac{1}{3} \) to have the same denominator.
\[ \frac{1}{3} \cdot \frac{12}{12} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} \]
Step 2.2.2: Perform the multiplication to find the equivalent fraction.
\[ \frac{12}{36} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} \]
Step 2.2.3: Scale \( \frac{1}{6} \) similarly.
\[ \frac{12}{36} + \frac{1}{6} \cdot \frac{6}{6} + \frac{1}{9} + \frac{1}{12} \]
Step 2.2.4: Multiply to find the equivalent fraction.
\[ \frac{12}{36} + \frac{6}{36} + \frac{1}{9} + \frac{1}{12} \]
Step 2.2.5: Scale \( \frac{1}{9} \) in the same way.
\[ \frac{12}{36} + \frac{6}{36} + \frac{1}{9} \cdot \frac{4}{4} + \frac{1}{12} \]
Step 2.2.6: Multiply to find the equivalent fraction.
\[ \frac{12}{36} + \frac{6}{36} + \frac{4}{36} + \frac{1}{12} \]
Step 2.2.7: Finally, scale \( \frac{1}{12} \).
\[ \frac{12}{36} + \frac{6}{36} + \frac{4}{36} + \frac{1}{12} \cdot \frac{3}{3} \]
Step 2.2.8: Multiply to find the equivalent fraction.
\[ \frac{12}{36} + \frac{6}{36} + \frac{4}{36} + \frac{3}{36} \]
Step 2.3: Add the numerators over the common denominator.
\[ \frac{12 + 6 + 4 + 3}{36} \]
Step 2.4: Combine the first two numbers.
\[ \frac{18 + 4 + 3}{36} \]
Step 2.5: Add the next number.
\[ \frac{22 + 3}{36} \]
Step 2.6: Add the final number.
\[ \frac{25}{36} \]
Step 3: Present the result in various forms.
Exact Form: \( \frac{25}{36} \) Decimal Form: \( \approx 0.6944 \)
Knowledge Notes:
The problem involves evaluating a finite summation, specifically the sum of the reciprocals of multiples of 3 from \( k = 1 \) to \( k = 4 \). The process of solving this problem includes expanding the summation into its individual terms, simplifying each term, finding a common denominator, and then combining the terms to find the sum.
Key knowledge points include:
Summation Notation: Understanding how to expand a summation into its individual terms.
Fraction Simplification: Knowing how to simplify fractions and find common denominators.
Arithmetic Operations: Performing addition of fractions, which requires a common denominator.
Exact vs. Decimal Form: Recognizing that the result of a summation can be expressed exactly as a fraction or approximately as a decimal.
The solution process demonstrates the step-by-step breakdown of each term in the series, the method to find a common denominator, and the arithmetic needed to combine the fractions into a single sum. The final answer is given in both exact form as a fraction and in decimal form for a practical approximation.
