Series with Geometric Terms
In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.
∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]

검증된 단계별 안내

Identify the general term of the series: \( a_n = \frac{1}{2^n} + \frac{(-1)^n}{5^n} \). This series is the sum of two separate geometric series.

Write out the first eight terms by substituting \( n = 0, 1, 2, \ldots, 7 \) into the general term: \( a_0, a_1, a_2, \ldots, a_7 \). This helps visualize how the series starts.

Recognize that each part is a geometric series: the first with common ratio \( r_1 = \frac{1}{2} \), and the second with common ratio \( r_2 = -\frac{1}{5} \). Both have initial terms \( a_0^{(1)} = 1 \) and \( a_0^{(2)} = 1 \) respectively.

Check the absolute values of the common ratios to determine convergence: since \( |r_1| < 1 \) and \( |r_2| < 1 \), both geometric series converge.

Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of each series separately, then add these sums to get the total sum of the original series.

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10m

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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 ∑ ar^n, where a is the first term and r is the common ratio. Understanding this helps in identifying and summing series with terms like (1/2)^n or (-1/5)^n.

Sum of an Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. Its sum is given by S = a / (1 - r). This formula is essential for finding the sum of each geometric component in the series when it converges.

Series Divergence and Convergence

A series converges if its partial sums approach a finite limit; otherwise, it diverges. For series composed of multiple parts, each part must be analyzed separately. Recognizing divergence or convergence is key to determining whether the sum exists or not.

Series with Geometric TermsIn Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]