Finding nth Partial Sums
In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.
2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

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Identify the type of series given. Notice that the series is 2 + \(\frac{2}{3}\) + \(\frac{2}{9}\) + \(\frac{2}{27}\) + \(\dots\), which is a geometric series where each term is multiplied by \(\frac{1}{3}\) to get the next term.

Write the general term of the series. The first term a is 2, and the common ratio r is \(\frac{1}{3}\). So, the nth term can be expressed as a_n = 2 \(\times\) \(\left\)(\(\frac{1}{3}\)\(\right\))^{n-1}.

Use the formula for the nth partial sum of a geometric series: S_n = a \(\times\) \(\frac{1 - r^n}{1 - r}\). Substitute a = 2 and r = \(\frac{1}{3}\) into this formula to get S_n.

Simplify the expression for S_n by plugging in the values and rewriting the formula clearly: S_n = 2 \(\times\) \(\frac{1 - \left(\frac{1}{3}\)\(\right\))^n}{1 - \(\frac{1}{3}\)}.

Determine if the series converges by examining the limit of S_n as n approaches infinity. Since |r| = \(\frac{1}{3}\) < 1, the series converges, and the sum to infinity is S = \(\frac{a}{1 - r}\).

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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. In this problem, the series has a common ratio of 1/3, making it geometric. Understanding this helps in finding a general formula for the nth partial sum.

Nth Partial Sum Formula

The nth partial sum of a geometric series is given by S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula allows calculation of the sum of the first n terms, which is essential for analyzing convergence and total sums.

Convergence of Infinite Series

An infinite geometric series converges if the absolute value of the common ratio is less than 1. When it converges, the sum approaches S = a / (1 - r) as n approaches infinity. Recognizing convergence is key to determining the series’ total sum.

Finding nth Partial SumsIn Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …