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

Options

A. `(1)/(3)[n-(1)/(9)(1-(1)/(100^(n)))]`
B. `(1)/(3)[n-(1)/(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)))]`

Answer

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

### Step 1: Express the terms in a more manageable form
The series can be rewritten as:
\[ 0.3 + 0.33 + 0.333 + \ldots = \frac{3}{10} + \frac{33}{100} + \frac{333}{1000} + \ldots \]
We can observe that each term can be expressed as:
\[ \frac{3}{10} + \frac{3 \times 11}{100} + \frac{3 \times 111}{1000} + \ldots \]

### Step 2: Rewrite the terms in a general form
Each term can be represented as:
\[ \frac{3 \times (10^k - 1)}{9 \times 10^k} \quad \text{for } k = 1, 2, \ldots, n \]
This simplifies to:
\[ \frac{3}{9} \left( \frac{10^k - 1}{10^k} \right) = \frac{1}{3} \left( 1 - \frac{1}{10^k} \right) \]

### Step 3: Sum the series
Now, we can express the sum of the series up to \(n\) terms:
\[ S_n = \sum_{k=1}^{n} \left( \frac{1}{3} \left( 1 - \frac{1}{10^k} \right) \right) \]
This can be separated into two sums:
\[ S_n = \frac{1}{3} \left( n - \sum_{k=1}^{n} \frac{1}{10^k} \right) \]

### Step 4: Calculate the geometric series
The sum \(\sum_{k=1}^{n} \frac{1}{10^k}\) is a geometric series with first term \(a = \frac{1}{10}\) and common ratio \(r = \frac{1}{10}\):
\[ \sum_{k=1}^{n} \frac{1}{10^k} = \frac{\frac{1}{10}(1 - \left(\frac{1}{10}\right)^n)}{1 - \frac{1}{10}} = \frac{1}{10} \cdot \frac{1 - \frac{1}{10^n}}{\frac{9}{10}} = \frac{1 - \frac{1}{10^n}}{9} \]

### Step 5: Substitute back into the sum
Substituting this back into our expression for \(S_n\):
\[ S_n = \frac{1}{3} \left( n - \frac{1 - \frac{1}{10^n}}{9} \right) \]
This simplifies to:
\[ S_n = \frac{1}{3} n - \frac{1}{27} + \frac{1}{27 \cdot 10^n} \]

### Final Result
Thus, the sum of the series \(0.3 + 0.33 + 0.333 + \ldots\) up to \(n\) terms is:
\[ S_n = \frac{1}{3} n - \frac{1}{27} + \frac{1}{27 \cdot 10^n} \]