Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)


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Verified step by step guidance

1. Recognize that the given series is \( \sum_{k=0}^{\infty} e^{-kx} \). This is a geometric series where each term can be written as \( (e^{-x})^k \).
2. Recall the formula for the sum of an infinite geometric series: \( \sum_{k=0}^{\infty} r^k = \frac{1}{1-r} \), which converges if and only if \( |r| < 1 \).
3. Identify the common ratio \( r = e^{-x} \) in this series. To apply the formula, we need to find the values of \( x \) such that \( |e^{-x}| < 1 \).
4. Since \( e^{-x} > 0 \) for all real \( x \), the inequality \( e^{-x} < 1 \) simplifies to \( -x < 0 \), or \( x > 0 \). This gives the interval of convergence.
5. Write the function represented by the series as \( f(x) = \frac{1}{1 - e^{-x}} \) for \( x > 0 \).

Key Concepts

Geometric Series and Its Sum

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. The sum of an infinite geometric series with first term a and common ratio r (|r| < 1) is a/(1 - r). Recognizing the given series as geometric allows us to express it as a closed-form function.

Interval of Convergence

The interval of convergence is the set of values for the variable x for which the infinite series converges. For geometric series, convergence requires the absolute value of the common ratio to be less than one. Determining this interval ensures the function representation is valid within that domain.

Exponential Functions and Their Properties

Exponential functions of the form e^{kx} have properties that simplify series expressions, especially when combined with geometric series concepts. Understanding how e^{-kx} behaves as k varies helps in identifying the ratio and analyzing convergence.