The series in the image is as follows:

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

This is a repeating decimal series. To find the sum, we can rewrite it as:

$S = 1 + 0.3 + 0.06 + 0.008 + 0.0003 + 0.00006 + 0.000008 + \dots$

The series alternates in groups of terms with a specific repeating pattern: $3, 6, 8$.

Step 1: Break the series into groups

The series can be written as:

$S = 1 + \left(0.3 + 0.06 + 0.008\right) + \left(0.0003 + 0.00006 + 0.000008\right) + \dots$

Each group consists of three terms. Factoring this repeating structure can simplify the calculation.

Step 2: Write the repeating part as a geometric series

Within each group, the terms form a geometric series. For the first group:

$G_1 = 0.3 + 0.06 + 0.008$

For the second group:

$G_2 = 0.0003 + 0.00006 + 0.000008$

The series progresses in powers of $10^{-3}$. Thus, the sum of the infinite series can be written as:

$S = 1 + \left(0.3 + 0.06 + 0.008\right) \cdot \left(1 + \frac{1}{10^3} + \frac{1}{10^6} + \dots\right)$

Step 3: Simplify each geometric sum

The sum of the first group is:

$0.3 + 0.06 + 0.008 = 0.368$

The factor $1 + \frac{1}{10^3} + \frac{1}{10^6} + \dots$ is an infinite geometric series with ratio $\frac{1}{10^3}$. Its sum is:

$\text{Sum of the infinite geometric series} = \frac{1}{1 - \frac{1}{10^3}} = \frac{1}{1 - 0.001} = \frac{1}{0.999} \approx 1.001$

Step 4: Combine everything

The total sum of the series is:

$S = 1 + 0.368 \cdot 1.001$

Simplify further:

$S \approx 1 + 0.368 \cdot 1.001 \approx 1 + 0.368 = 1.368$

Final Answer:

$\boxed{1.368}$

Would you like a detailed breakdown of any step or additional explanations?

Related Questions:

1. What is the general formula for a geometric series and how is it applied?
2. How does the repeating decimal pattern affect the convergence of the series?
3. What are other methods to approach infinite series with repeating patterns?
4. Can this series be converted into a fraction for an exact representation?
5. How do small perturbations (like $\pm 0.001$) impact calculations in such series?

Tip:

Understanding the geometric series formula is essential for working with infinite repeating patterns in mathematics.