Find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

검증된 단계별 안내

Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( a = 1 \).

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

Check if the series converges by verifying that the absolute value of the common ratio is less than 1: \( |r| = \left| -\frac{1}{2} \right| = \frac{1}{2} < 1 \), so the series converges.

Use the formula for the sum \( S \) of an infinite geometric series: \[ S = \frac{a}{1 - r} \].

Substitute the values of \( a \) and \( r \) into the formula to express the sum: \[ S = \frac{1}{1 - \left(-\frac{1}{2}\right)} \].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. The series continues indefinitely, and its behavior depends on the common ratio's value.

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. For convergence of an infinite series, the absolute value of the common ratio must be less than 1.

Sum Formula for Infinite Geometric Series

If the absolute value of the common ratio r is less than 1, the sum S of the infinite geometric series with first term a is given by S = a / (1 - r). This formula allows calculation of the series' sum without adding infinitely many terms.