In Exercises 37–44, find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

Verified step by step guidance

1. Identify the first term \( a \) of the series, which is 1.
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-1/2}{1} = -\frac{1}{2} \).
3. Check if the series is convergent by ensuring that the absolute value of the common ratio is less than 1: \( |r| = |-\frac{1}{2}| = \frac{1}{2} < 1 \).
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 \( a \) and \( r \) into the formula to find the sum: \( S = \frac{1}{1 - (-\frac{1}{2})} \).

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of an infinite number of terms where each term after the first is found by multiplying the previous term by a constant called the common ratio. The series converges to a finite value if the absolute value of the common ratio is less than one.

Sum of an Infinite Geometric Series

The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio. This formula is applicable only when the absolute value of 'r' is less than one, ensuring convergence.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. In the series 1 - 1/2 + 1/4 - 1/8 + ..., the common ratio is -1/2, indicating that each term is obtained by multiplying the previous term by -1/2.