Find the sum of each infinite geometric series. 3 + 3/4 + 3/4² + 3/4³ + ...

Verified step by step guidance

1. Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( 3 \).
2. Determine the common ratio \( r \) by dividing the second term by the first term. Here, \( r = \frac{3/4}{3} = \frac{1}{4} \).
3. Verify that the absolute value of the common ratio \( |r| < 1 \) to ensure the series converges. Since \( \frac{1}{4} < 1 \), the series converges.
4. Use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio.
5. Substitute the values of \( a = 3 \) and \( r = \frac{1}{4} \) into the formula to express the sum \( S \) without calculating the final value.

Key Concepts

Infinite Geometric Series

An infinite geometric series is the sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. It has the form a + ar + ar² + ar³ + ..., where 'a' is the first term and 'r' is the common ratio.

Common Ratio

The common ratio 'r' in a geometric series is the factor by which each term is multiplied to get the next term. It is found by dividing any term by its preceding term. For the series to converge, the absolute value of 'r' must be less than 1.

Sum of an Infinite Geometric Series

If the absolute value of the common ratio is less than 1, the infinite geometric series converges, and its sum can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.