Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...

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Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( 2 \).

Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-1}{2} \).

Check if the series converges by verifying that the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \).

Use the formula for the sum of an infinite geometric series, which is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

Substitute the values of \( a \) and \( r \) into the formula to express the sum \( S \) without calculating the final numerical value.

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주요 개념

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Infinite Geometric Series

An infinite geometric series is a sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. The series continues indefinitely, and its behavior depends on the common ratio.

Common Ratio

The common ratio 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 and is crucial for determining the series' convergence.

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.