Find the sum of an infinite geometric series: 1 + 1/3 + 1/9 + 1/27 + ...

Step 1: Identify the first term and common ratio

In a geometric series, the first term is denoted by a, and the common ratio is denoted by r. Here, the first term a is 1. The common ratio r is found by dividing any term by its preceding term, for example, $\frac{1/3}{1} = \frac{1}{3}$ or $\frac{1/9}{1/3} = \frac{1}{3}$.

Step 2: Check condition for sum of infinite geometric series

The sum of an infinite geometric series converges (exists) if and only if the absolute value of the common ratio r is less than 1. In this case, | $\frac{1}{3}| = \frac{1}{3}$, which is less than 1, so the sum exists.

Step 3: Apply the formula for the sum of an infinite geometric series

The formula for the sum S of an infinite geometric series with first term a and common ratio r (where |r| < 1) is $S = \frac{a}{1 - r}$.

Step 4: Substitute values and calculate the sum

Substitute the values $a = 1$ and $r = \frac{1}{3}$ into the formula. Simplify the expression to find the sum S. First, subtract $\frac{1}{3}$ from 1 in the denominator, which gives $\frac{2}{3}$. Then, divide 1 by $\frac{2}{3}$, which is equivalent to multiplying 1 by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$.

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