Evaluate the Summation sum from k=1 to infinity of 7^(-k)

Solution:

Step 1:

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

Step 2:

Calculate the common ratio using the formula $r = \frac{a_{k+1}}{a_k}$.

Step 2.1:

Insert $a_k$ and $a_{k+1}$ into the ratio formula: $r = \frac{7^{-(k+1)}}{7^{-k}}$.

Step 2.2:

Proceed to simplify the expression.

Step 2.2.1:

Eliminate the common base of $7^{-k}$ and $7^{-(k+1)}$.

Step 2.2.1.1:

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

Step 2.2.1.2:

Remove the common factors.

Step 2.2.1.2.1:

Multiply by unity: $r = \frac{7^{-k} \cdot 7^{-(k+1)+k}}{7^{-k} \cdot 1}$.

Step 2.2.1.2.2:

Cancel out the common base: $r = \frac{\cancel{7^{-k}} \cdot 7^{-(k+1)+k}}{\cancel{7^{-k}} \cdot 1}$.

Step 2.2.1.2.3:

Reformulate the expression: $r = \frac{7^{-(k+1)+k}}{1}$.

Step 2.2.1.2.4:

Divide by one: $r = 7^{-(k+1)+k}$.

Step 2.2.2:

Simplify the exponent.

Step 2.2.2.1:

Apply the power rule: $r = 7^{-k-1+k}$.

Step 2.2.2.2:

Simplify the exponent further: $r = 7^{-1}$.

Step 2.2.3:

Combine like terms in the exponent.

Step 2.2.3.1:

Add together: $r = 7^{0-1}$.

Step 2.2.3.2:

Subtract to simplify: $r = 7^{-1}$.

Step 2.2.4:

Apply the rule for negative exponents: $r = \frac{1}{7}$.

Step 3:

Verify the convergence of the series since $|r| < 1$.

Step 4:

Identify the first term of the series by substituting $k = 1$.

Step 4.1:

Replace $k$ with $1$ in $7^{-k}$: $a = 7^{-1}$.

Step 4.2:

Use the negative exponent rule: $a = \frac{1}{7}$.

Step 5:

Insert the values of $a$ and $r$ into the sum formula: $S = \frac{\frac{1}{7}}{1 - \frac{1}{7}}$.

Step 6:

Simplify the expression.

Step 6.1:

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

Step 6.2:

Simplify the denominator.

Step 6.2.1:

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

Step 6.2.2:

Combine the fractions over a common denominator: $S = \frac{1}{7} \cdot \frac{1}{\frac{6}{7}}$.

Step 6.2.3:

Subtract within the denominator: $S = \frac{1}{7} \cdot \frac{7}{6}$.

Step 6.3:

Multiply the reciprocal of the denominator by the numerator: $S = \frac{1}{7} \cdot \frac{7}{6}$.

Step 6.4:

Multiply through: $S = \frac{1}{7} \cdot \frac{7}{6}$.

Step 6.5:

Eliminate the common factor.

Step 6.5.1:

Cancel out the 7: $S = \frac{1}{\cancel{7}} \cdot \frac{\cancel{7}}{6}$.

Step 6.5.2:

Finalize the expression: $S = \frac{1}{6}$.

Step 7:

Present the final result in various formats.

Exact Form: $\frac{1}{6}$ Decimal Form: $0.1666...$

Knowledge Notes:

1. Infinite Geometric Series: An infinite geometric series is a series of the form $\sum_{k=1}^{\infty} ar^{k-1}$, where $a$ is the first term and $r$ is the common ratio. If $|r| < 1$, the series converges to $\frac{a}{1 - r}$.

2. Convergence Criteria: For an infinite geometric series to converge, the absolute value of the common ratio must be less than one, i.e., $|r| < 1$.

3. Common Ratio: The common ratio $r$ in a geometric series is the factor by which each term is multiplied to get the next term. It can be found by dividing any term by the preceding term.

4. Negative Exponent Rule: The negative exponent rule states that $b^{-n} = \frac{1}{b^n}$, which is used to simplify expressions with negative exponents.

5. Simplifying Fractions: When simplifying expressions involving fractions, finding a common denominator and canceling out common factors can help to simplify the calculation.

6. Multiplying by Reciprocal: To divide by a fraction, you multiply by its reciprocal. This is used when simplifying complex fraction expressions.