21–42. Geometric series Evaluate each geometric series or state that it diverges.

37.1 + e/π + e²/π² + e³/π³ + ⋯

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Identify the first term \( a \) of the geometric series. Here, the first term is \( 1 \).

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

Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| \frac{e}{\pi} \right| \). The series converges if \( |r| < 1 \) and diverges otherwise.

If the series converges, 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.

Substitute the values of \( a \) and \( r \) into the formula to express the sum of the series.

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주요 개념

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

Geometric Series

A geometric series is a sum of 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.

Convergence of Geometric Series

A geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges and does not have a finite sum.

Evaluating the Given Series

To evaluate the series 1 + e/π + e²/π² + ⋯, identify the first term a = 1 and the common ratio r = e/π. Since e ≈ 2.718 and π ≈ 3.1415, |r| < 1, so the series converges. Use the sum formula S = 1 / (1 - e/π) to find the sum.

21–42. Geometric series Evaluate each geometric series or state that it diverges. 37.1 + e/π + e²/π² + e³/π³ + ⋯