Find the sum of an infinite geometric series: 1 + 1/3 + 1/9 + 1/27 + ...
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Step-by-Step Solution
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}$.