The given series appears to follow a repetitive pattern in its digits and denominators. Let’s carefully analyze and calculate its sum.

The series is:

$1 + \frac{1}{10} + \frac{4}{10^2} + \frac{6}{10^3} + \frac{1}{10^4} + \frac{4}{10^5} + \frac{6}{10^6} + \frac{1}{10^7} + \frac{4}{10^8} + \frac{6}{10^9} + \dots$

This is an infinite series with repeating digits $1, 4, 6$ in the decimal places. Let’s write it as:

$S = 1 + 0.1 + 0.004 + 0.0006 + 0.0001 + 0.00004 + 0.000006 + \dots$

Step 1: Recognizing the repeating pattern

The digits $1, 4, 6$ repeat every 3 terms. Hence, the series can be grouped as:

$S = (1 + 0.0001 + 0.0000001 + \dots) + (0.1 + 0.00004 + 0.000000004 + \dots) + (0.004 + 0.000006 + 0.000000006 + \dots)$

Each group is a geometric series.

Step 2: Sum of each geometric series

For the first group ($1 + 0.0001 + 0.0000001 + \dots$):

• First term ($a$) = $1$
• Common ratio ($r$) = $0.0001$

The sum of an infinite geometric series is:

$\text{Sum} = \frac{a}{1 - r} = \frac{1}{1 - 0.0001} = \frac{1}{0.9999} \approx 1.0001$

For the second group ($0.1 + 0.00004 + 0.000000004 + \dots$):

• First term ($a$) = $0.1$
• Common ratio ($r$) = $0.0004$

$\text{Sum} = \frac{a}{1 - r} = \frac{0.1}{1 - 0.0004} = \frac{0.1}{0.9996} \approx 0.10004$

For the third group ($0.004 + 0.000006 + 0.000000006 + \dots$):

• First term ($a$) = $0.004$
• Common ratio ($r$) = $0.000001$

$\text{Sum} = \frac{a}{1 - r} = \frac{0.004}{1 - 0.000001} \approx 0.004$

Step 3: Adding the results

Now, add the sums of the three groups:

$S \approx 1.0001 + 0.10004 + 0.004 \approx 1.10414$

Final Answer:

The sum of the infinite series is approximately:

$S = 1.10414$

Would you like me to elaborate further or verify any part? Here are some additional questions to explore:

1. How does the sum change if the repeating pattern changes?
2. What happens if we truncate the series after a certain number of terms?
3. How does the calculation change for a different common ratio?
4. Can this series be expressed as a single decimal expansion?
5. How would the result vary if the sequence of repeating digits differs?

Tip: When dealing with repeating decimals or series, always check for geometric patterns to simplify calculations.