The given series is:

$1 + \frac{2}{10} + \frac{7}{10^2} + \frac{6}{10^3} + \frac{2}{10^4} + \frac{7}{10^5} + \frac{6}{10^6} + \cdots$

This series has a repeating pattern: $1, 2, 7, 6$. Let's express it as a decimal.

Step 1: Represent the series as a decimal

Each group of four digits $1, 2, 7, 6$ repeats. The series is equivalent to:

$1.276276276\ldots$

This is a repeating decimal with a block of $276$ repeating after the initial digit $1$.

Step 2: Convert to a fraction

To convert $1.276276276\ldots$ into a fraction:

1. Let $x = 1.276276276\ldots$
2. Multiply by $10^3$ (since the repeating block has 3 digits):

$1000x = 1276.276276276\ldots$

3. Subtract the original $x$:

$1000x - x = 1276.276276276\ldots - 1.276276276\ldots$

$999x = 1275$

$x = \frac{1275}{999}$

Step 3: Simplify the fraction

Simplify $\frac{1275}{999}$ by dividing the numerator and denominator by their greatest common divisor (GCD), which is $3$:

$x = \frac{1275 \div 3}{999 \div 3} = \frac{425}{333}$

Final Answer:

The value of the series is:

$\boxed{\frac{425}{333}}$

Let me know if you'd like further clarification or additional details! Here are five related questions to explore this concept:

1. How can you identify the repeating block in a decimal expansion?
2. What is the method to convert a repeating decimal to a fraction for any repeating block size?
3. How does this series relate to geometric series?
4. Can we calculate the sum using an algebraic formula directly?
5. What are some practical applications of repeating decimals in mathematics?

Tip: To convert repeating decimals quickly, focus on identifying the repeating block and use the formula for fractions!