What is the sum of the infinite geometric series: 3 - 3/2 + 3/4 - 3/8 + ...?

To find the sum of the infinite geometric series given by (3 - \frac{3}{2} + \frac{3}{4} - \frac{3}{8} + \ldots), we first need to identify the first term and the common ratio.

The first term of the series, denoted as (a), is 3. Next, we need to determine the common ratio, (r). We can find (r) by dividing the second term by the first term:

[

r = -\frac{3/2}{3} = -\frac{1}{2}

]

This ratio indicates that each term is obtained by multiplying the previous term by (-\frac{1}{2}).

For an infinite geometric series, the sum can be calculated using the formula:

[

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

]

This formula is valid when the absolute value of the common ratio is less than 1, which is the case here since (|r| = \frac{1}{2} < 1).

Substituting the values we found into the formula:

[

S = \frac{3}{1 - \left