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


29.∑ (k = 1 to ∞) e^(–2k)

Verified step by step guidance

1. Identify the general term of the series. Here, the series is given by \( \sum_{k=1}^{\infty} e^{-2k} \). The general term \( a_k \) is \( e^{-2k} \).
2. Recognize that this is a geometric series because each term can be written as \( (e^{-2})^k \), where the common ratio \( r = e^{-2} \).
3. Recall the convergence criterion for an infinite geometric series: it converges if and only if \( |r| < 1 \). Since \( e^{-2} \) is positive and less than 1, the series converges.
4. Use the formula for the sum of an infinite geometric series starting at \( k=1 \): \[ S = \frac{a_1}{1 - r} \], where \( a_1 = e^{-2} \) and \( r = e^{-2} \).
5. Substitute the values into the formula to express the sum as \[ S = \frac{e^{-2}}{1 - e^{-2}} \]. This expression represents the sum of the series.

Key Concepts

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^(k), where a is the first term and r is the common ratio. Understanding this structure helps identify and evaluate the series.

Convergence of Infinite Geometric Series

An infinite 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.

Exponential Functions in Series

Exponential terms like e^(–2k) can be rewritten to identify the common ratio in a geometric series. Recognizing that e^(–2k) = (e^(–2))^k allows us to treat the series as geometric with ratio r = e^(–2), facilitating convergence tests and sum calculations.