What is the sum of the series
`0.3+0.33+0.333+....n` terms?

Options

A. `(1)/(3)[n-(1)/(9)(1-(1)/(10^(n)))]`
B. `(1)/(3)[n-(2)/(9)(1-(1)/(10^(n)))]`
C. `(1)/(3)[n-(1)/(3)(1-(1)/(10^(n)))]`
D. `(1)/(3)[n-(1)/(9)(1+(1)/(10^(n)))]`

Explanation

To find the sum of the series \( S_n = 0.3 + 0.33 + 0.333 + \ldots \) up to \( n \) terms, we can follow these steps:

### Step 1: Rewrite the terms in a more manageable form
Each term in the series can be expressed as:
- \( 0.3 = \frac{3}{10} \)
- \( 0.33 = \frac{33}{100} = \frac{3 \cdot 11}{100} \)
- \( 0.333 = \frac{333}{1000} = \frac{3 \cdot 111}{1000} \)

We can see that each term can be expressed as:
\[ \text{Term}_k = \frac{3 \cdots 3}{10^k} \]
where \( k \) is the number of digits '3' in the numerator.

### Step 2: Factor out the common term
Notice that each term has a common factor of \( 3 \):
\[ S_n = 3 \left( \frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + \ldots \right) \]

### Step 3: Express the series in a summable form
The series inside the parentheses can be rewritten as:
\[ S_n = 3 \left( \frac{1}{10} + \frac{1 \cdot 10 + 1}{100} + \frac{1 \cdot 100 + 1 \cdot 10 + 1}{1000} + \ldots \right) \]

### Step 4: Recognize the pattern
The series can be expressed as:
\[ S_n = 3 \left( \sum_{k=1}^{n} \frac{10^k - 1}{9 \cdot 10^k} \right) \]

### Step 5: Simplify the series
Using the formula for the sum of a geometric series, we can simplify the series:
\[ S_n = 3 \cdot \frac{1}{9} \left( \frac{10}{9} \left( 1 - \left( \frac{1}{10} \right)^n \right) \right) \]

### Step 6: Final expression
Putting it all together, we have:
\[ S_n = \frac{1}{3} n - \frac{1}{9} \left( 1 - \left( \frac{1}{10} \right)^n \right) \]

### Final Result
Thus, the sum of the series \( S_n \) up to \( n \) terms is:
\[ S_n = \frac{1}{3} n - \frac{1}{9} \left( 1 - \left( \frac{1}{10} \right)^n \right) \]