Evaluate the Summation sum from i=1 to 4 of 1/(3i)
The question is asking for the calculation of a finite mathematical series. Specifically, it wants the sum of the reciprocals of three times the index variable 'i', starting with 'i' equal to 1 and ending with 'i' equal to 4. The series is thus composed of four terms, each term being of the form 1/(3i) where 'i' takes on the integer values from 1 to 4, inclusively. To evaluate the summation, one would calculate each term individually and then add them together to find the total sum.
$\sum_{i = 1}^{4} \frac{1}{3 i}$
Answer
Solution:
Step 1: Write out the series for the given range of $i$.
$\frac{1}{3 \cdot 1} + \frac{1}{3 \cdot 2} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4}$
Step 2: Begin simplification.
Step 2.1: Simplify each fraction.
Step 2.1.1: Remove the common factor of $1$.
$\frac{\cancel{1}}{3 \cdot \cancel{1}} + \frac{1}{3 \cdot 2} + \frac{1}{3 \cdot 3} + \frac{1}{3 \cdot 4}$
Step 2.1.2: Rewrite the simplified terms.
$\frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12}$
Step 2.2: Convert to a common denominator.
Step 2.2.1: Multiply $\frac{1}{3}$ by $\frac{4}{4}$.
$\frac{4}{12} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12}$
Step 2.2.2: Multiply $\frac{1}{6}$ by $\frac{2}{2}$.
$\frac{4}{12} + \frac{2}{12} + \frac{1}{9} + \frac{1}{12}$
Step 2.2.3: Multiply $\frac{1}{9}$ by $\frac{4}{4}$.
$\frac{4}{12} + \frac{2}{12} + \frac{4}{36} + \frac{1}{12}$
Step 2.2.4: Combine the fractions with a common denominator.
$\frac{4 + 2 + 1}{12} + \frac{4}{36}$
Step 2.2.5: Simplify the combined fraction.
$\frac{7}{12} + \frac{4}{36}$
Step 2.2.6: Multiply $\frac{7}{12}$ by $\frac{3}{3}$ to get the common denominator.
$\frac{21}{36} + \frac{4}{36}$
Step 2.3: Add the fractions with the common denominator.
$\frac{21 + 4}{36}$
Step 2.4: Simplify the final sum.
$\frac{25}{36}$
Step 3: Present the result in various forms.
Exact Form: $\frac{25}{36}$ Decimal Form: $0.\overline{694444}$
Knowledge Notes:
The problem involves evaluating a finite summation of a series where the general term is given by $\frac{1}{3i}$. The steps taken in the solution involve:
Expansion of the Series: The summation is expanded by substituting the values of $i$ from $1$ to $4$ into the general term.
Simplification of Terms: Each term of the series is simplified by performing basic arithmetic operations. Since the terms are fractions, simplification may involve reducing the fractions to their simplest form.
Finding a Common Denominator: To add fractions, they must have a common denominator. The least common multiple (LCM) of the denominators is often used as the common denominator.
Combining Fractions: Once a common denominator is found, the numerators of the fractions are added while the common denominator remains the same.
Final Summation: The final step is to combine the numerators over the common denominator and simplify the result if possible.
Representation of the Result: The result can be expressed in its exact form (as a fraction) or converted to a decimal form. The decimal form may be a terminating or repeating decimal, depending on the fraction.
Understanding these steps is crucial for solving similar problems involving summation of series and manipulation of fractions.
