Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)

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Identify the type of series given: \( \sum_{n=0}^{\infty} e^{-2n} \). Notice that this is a geometric series because each term can be written as \( (e^{-2})^n \).

Recall the formula for the sum of an infinite geometric series: if \( |r| < 1 \), then \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio.

Determine the first term \( a \) and the common ratio \( r \) for this series. Here, \( a = e^{-2 \cdot 0} = 1 \) and \( r = e^{-2} \).

Check the convergence condition by verifying if \( |r| < 1 \). Since \( e^{-2} \) is a positive number less than 1, the series converges.

Use the sum formula to express the sum of the series as \( \frac{1}{1 - e^{-2}} \). This represents the sum of the infinite series.

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주요 개념

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Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio r. It converges if the absolute value of r is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term.

Convergence Criteria for Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. For geometric series, this depends on the common ratio, while for other series, tests like the comparison or ratio test may be used to determine convergence or divergence.

Exponential Functions in Series

Exponential functions of the form e^(−kn) decrease rapidly as n increases when k > 0. When used as terms in a series, they often form geometric series with ratio e^(−k), which helps in analyzing convergence and calculating sums.

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.∑ (from n = 0 to ∞) e^(−2n)