Evaluate the Summation sum from i=1 to infinity of (4/7)^i

Solution:

Step 1:

To determine the sum of an infinite geometric series, use the formula $S = \frac{a}{1 - r}$, where $S$ is the sum, $a$ is the initial term, and $r$ is the common ratio.

Step 2:

Identify the common ratio using the formula $r = \frac{a_{i+1}}{a_i}$.

Step 2.1:

Insert the terms $a_i$ and $a_{i+1}$ into the ratio formula: $r = \frac{(\frac{4}{7})^{i+1}}{(\frac{4}{7})^i}$.

Step 2.2:

Eliminate the shared term in both the numerator and the denominator.

Step 2.2.1:

Extract $(\frac{4}{7})^i$ from $(\frac{4}{7})^{i+1}$: $r = \frac{(\frac{4}{7})^i \cdot \frac{4}{7}}{(\frac{4}{7})^i}$.

Step 2.2.2:

Remove the common term.

Step 2.2.2.1:

Multiply by the identity element (1): $r = \frac{(\frac{4}{7})^i \cdot \frac{4}{7}}{(\frac{4}{7})^i \cdot 1}$.

Step 2.2.2.2:

Cancel out the common term: $r = \frac{\cancel{(\frac{4}{7})^i} \cdot \frac{4}{7}}{\cancel{(\frac{4}{7})^i} \cdot 1}$.

Step 2.2.2.3:

Simplify the expression: $r = \frac{\frac{4}{7}}{1}$.

Step 2.2.2.4:

Solve for $r$: $r = \frac{4}{7}$.

Step 3:

Confirm that the series converges since $|r| < 1$.

Step 4:

Calculate the first term of the series by setting $i = 1$.

Step 4.1:

Replace $i$ with 1 in the term: $a = (\frac{4}{7})^1$.

Step 4.2:

Simplify to find $a$: $a = \frac{4}{7}$.

Step 5:

Insert $a$ and $r$ into the summation formula: $S = \frac{\frac{4}{7}}{1 - \frac{4}{7}}$.

Step 6:

Proceed to simplify the expression.

Step 6.1:

Multiply the numerator by the reciprocal of the denominator: $S = \frac{4}{7} \cdot \frac{1}{1 - \frac{4}{7}}$.

Step 6.2:

Simplify the denominator.

Step 6.2.1:

Express 1 as a fraction with a common denominator: $S = \frac{4}{7} \cdot \frac{1}{\frac{7}{7} - \frac{4}{7}}$.

Step 6.2.2:

Combine the fractions over the common denominator: $S = \frac{4}{7} \cdot \frac{1}{\frac{7 - 4}{7}}$.

Step 6.2.3:

Subtract within the denominator: $S = \frac{4}{7} \cdot \frac{1}{\frac{3}{7}}$.

Step 6.3:

Multiply by the reciprocal of the denominator: $S = \frac{4}{7} \cdot \frac{7}{3}$.

Step 6.4:

Multiply the fractions: $S = \frac{4}{7} \cdot \frac{7}{3}$.

Step 6.5:

Cancel out the common factor.

Step 6.5.1:

Eliminate the common factor of 7: $S = \frac{4}{\cancel{7}} \cdot \frac{\cancel{7}}{3}$.

Step 6.5.2:

Express the simplified form: $S = 4 \cdot \frac{1}{3}$.

Step 6.6:

Combine the terms to find the sum: $S = \frac{4}{3}$.

Step 7:

The summation result can be presented in various formats:

• Exact Form: $\frac{4}{3}$
• Decimal Form: $1.333...$
• Mixed Number Form: $1 \frac{1}{3}$

Knowledge Notes:

To solve the problem of evaluating the sum of an infinite geometric series, one must understand several key concepts:

1. Infinite Geometric Series: A series with a constant ratio between successive terms. It can be summed up if the absolute value of the common ratio $|r|$ is less than 1.

2. Common Ratio ($r$): The factor by which each term of the series is multiplied to obtain the next term. It is found by dividing any term by the preceding term.

3. First Term ($a$): The initial term of the series, which is critical for calculating the sum.

4. Summation Formula: For an infinite geometric series, the sum is given by $S = \frac{a}{1 - r}$, where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio.

5. Convergence: An infinite series converges if the sum approaches a finite value as the number of terms increases indefinitely. For a geometric series, this occurs when $|r| < 1$.

6. Simplification: The process of reducing an expression to its simplest form, often by canceling common factors or combining like terms.

7. Reciprocal: The reciprocal of a number is 1 divided by that number. It is used in the process of division, especially when simplifying complex fractions.

8. Representation of Results: The sum of an infinite series can be expressed in various forms, including exact fractions, decimals, or mixed numbers. The choice of representation depends on the context and the desired precision.