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 ,11/3=31or1/31/9=31.
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 ,∣31∣=31, 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=1−ra.
Step 4: Substitute values and calculate the sum
Substitute the values a=1 and r=31 into the formula. Simplify the expression to find the sum S .First,subtract31from1 in the denominator, which gives 32. Then, divide 1 by 32 which is equivalent to multiplying 1 by the reciprocal of 32, which is 23.