Evaluate the Summation sum from i=1 to 4 of 1/(3i)

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:

1. Expansion of the Series: The summation is expanded by substituting the values of $i$ from $1$ to $4$ into the general term.

2. 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.

3. 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.

4. Combining Fractions: Once a common denominator is found, the numerators of the fractions are added while the common denominator remains the same.

5. Final Summation: The final step is to combine the numerators over the common denominator and simplify the result if possible.

6. 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.