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

Verified step by step guidance

1. Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( 1 \).
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{1/3}{1} = \frac{1}{3} \).
3. Verify that the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \), which ensures the series converges. Here, \( |\frac{1}{3}| < 1 \), so the series converges.
4. Use the formula for the sum \( S \) 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 \( a = 1 \) and \( r = \frac{1}{3} \) into the formula to express the sum of the series.

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 |r| < 1 for the series to converge.

Common Ratio

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

Sum Formula for Infinite Geometric Series

The sum S of an infinite geometric series with first term a and common ratio r (|r| < 1) is given by S = a / (1 - r). This formula allows calculation of the total sum without adding infinitely many terms.