Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} e^{-2 k}\)

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty} e^{-2 k}\) converges.

Step 1: Identify the Ratio

The given series is \(\sum_{k=1}^{\infty} e^{-2 k}\), which is a geometric series. Comparing this with the general formula for a geometric series, we can see that our ratio r is \(e^{-2}\).

Step 2: Apply the Geometric Series Test

The Geometric Series Test states that a geometric series converges if the absolute value of the ratio is less than one, and diverges if the absolute value of the ratio is greater than or equal to one. The absolute value of our ratio, \(|e^{-2}|\), which is equal to \(e^{-2}\), since \(e^{-2}\) is always positive.

Step 3: Determine if the series converges or diverges

Since \(e^{-2} \approx 0.135 < 1\), according to the Geometric Series Test, the series \(\sum_{k=1}^{\infty} e^{-2 k}\) converges.

Key Concepts

Convergence and Divergence

Understanding whether a series converges or diverges is fundamental in mathematics, especially when dealing with infinite sums. When we speak of a series' convergence, we refer to the sum of its terms approaching a definite value as we consider more and more terms. Conversely, when a series diverges, the sum does not settle on a particular value, even as the number of terms grows indefinitely.

For example, the series \(\sum_{k=1}^{\infty} e^{-2k}\) is examined to see if the sum of its infinitely many terms approaches a specific limit. This determination is essential for various applications in fields such as engineering, physics, and finance, where infinite processes must be summed up to make predictions or calculate quantities.

Geometric Series Test

The Geometric Series Test is a specific criterion used to determine the convergence of geometric series. A geometric series is of the form \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio between consecutive terms. To use this test, identify the series' ratio and take its absolute value. If the absolute value is less than one (\(|r| < 1\)), the series converges; if it is greater than or equal to one, the series diverges.

In our example, identifying \(e^{-2}\) as the ratio and noting that it is less than one indicates that the series converges. It's a quick, efficient way to determine the behavior of a series and is particularly handy for series with exponential terms.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers, which can pose quite the conceptual challenge. While adding up a finite number of elements is straightforward, summing infinitely many requires the concept of a limit. The infinite series is said to converge if the partial sums approach a fixed number as the number of terms increases indefinitely.

The series in question, \(\sum_{k=1}^{\infty} e^{-2k}\), would have us adding up infinitely many terms of the form \(e^{-2k}\). Thanks to tests like the Geometric Series Test, we can conclude about the behavior of such infinite series without actually adding up an infinite number of terms, which is practically impossible.

Exponential Functions

Exponential functions are characterized by their constant bases raised to variable exponents and are expressed as \(f(x) = a^x\), where \(a\) is a constant. They are incredibly important in modeling growth and decay processes in various scientific disciplines. For instance, in our series \(\sum_{k=1}^{\infty} e^{-2k}\), the exponential function is \(e^{-2k}\), which represents a decay as \(k\) increases.

Because of the properties of exponential functions, where the base \(e\) is a mathematical constant approximately equal to 2.71828, they often appear in problems involving continuous growth or decay, reinforcing the relevance of understanding their behavior in infinite series.