What is the sum of `sqrt(3)+(1)/(sqrt(3))+(1)/(3sqrt(3))+......`?

Options

A. `(sqrt(3))/(2)`
B. `(3sqrt(3))/(2)`
C. `(2sqrt(3))/(3)`
D. `sqrt(3)`

Explanation

To find the sum of the series \( \sqrt{3} + \frac{1}{\sqrt{3}} + \frac{1}{3\sqrt{3}} + \ldots \), we can identify the pattern and determine if it forms a geometric series.

### Step 1: Identify the first term and common ratio
The first term \( a \) of the series is \( \sqrt{3} \).

Next, we need to find the common ratio \( r \). The second term is \( \frac{1}{\sqrt{3}} \), and we can find \( r \) by dividing the second term by the first term:

\[ r = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3} \]

### Step 2: Check if the series is a geometric series
To confirm that this is a geometric series, we can check if the ratio of consecutive terms is constant.

- The first term is \( a = \sqrt{3} \).
- The second term is \( \frac{1}{\sqrt{3}} \).
- The third term is \( \frac{1}{3\sqrt{3}} \).

Calculating the ratio of the second term to the first term:

\[ \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3} \]

Calculating the ratio of the third term to the second term:

\[ \frac{\frac{1}{3\sqrt{3}}}{\frac{1}{\sqrt{3}}} = \frac{1}{3} \]

Since both ratios are equal, the series is indeed a geometric series with first term \( a = \sqrt{3} \) and common ratio \( r = \frac{1}{3} \).

### Step 3: Use the formula for the sum of an infinite geometric series
The sum \( S \) of an infinite geometric series is given by the formula:

\[ S = \frac{a}{1 - r} \]

Substituting the values we found:

\[ S = \frac{\sqrt{3}}{1 - \frac{1}{3}} = \frac{\sqrt{3}}{\frac{2}{3}} = \sqrt{3} \cdot \frac{3}{2} = \frac{3\sqrt{3}}{2} \]

### Conclusion
The sum of the series \( \sqrt{3} + \frac{1}{\sqrt{3}} + \frac{1}{3\sqrt{3}} + \ldots \) is

\[ \frac{3\sqrt{3}}{2} \]