Problem 55 - Thomas Calculus

Question

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty} e^{-2 n} $$

Step-by-Step Solution

Answer

The series converges with sum \(\frac{1}{1-e^{-2}}\).

Step 1: Identify the Series Type

The series given is \(\sum_{n=0}^{\infty} e^{-2n}\). This is a geometric series where each term is in the form \(ar^n\) with \(a = 1\) (the first term \(e^{-2\cdot0} = 1\)) and \(r = e^{-2}\) (the common ratio).

Step 2: Determine the Common Ratio

The common ratio of the series is \(r = e^{-2}\). Since \(e^{-2} \approx 0.135\), we observe that \(0 < r < 1\).

Step 3: Apply Geometric Series Convergence Criterion

A geometric series \(\sum_{n=0}^{\infty} ar^n\) converges if the common ratio \(|r| < 1\). Here, \(|e^{-2}| < 1\), hence the series converges.

Step 4: Find the Sum of the Convergent Geometric Series

To find the sum of a convergent geometric series, we use the formula \(S = \frac{a}{1-r}\), where \(a = 1\) and \(r = e^{-2}\). Substituting in, we get \(S = \frac{1}{1-e^{-2}}\).

Key Concepts

Convergence of Series

When we speak of the convergence of a series, we're looking at whether the sum of the series approaches a finite number as more and more terms are added. Specifically for a geometric series \(\sum_{n=0}^{\infty} ar^n\), convergence occurs if the absolute value of the common ratio \(r\) is less than 1. In simpler terms, the terms become increasingly smaller and decrease rapidly enough that they accumulate to a fixed, finite sum.

This is crucial because some series might appear to grow ever larger and never settle at a number. Such series are said to "diverge," like \(1 + 1 + 1 + \ldots\). For a geometric series to converge, observe the common ratio: if \(|r| < 1\), you’re good to proceed. If not, the series will diverge.

Common Ratio

The common ratio \(r\) in a geometric series is the factor by which we multiply one term to get the next. It defines how quickly the terms change. Understanding the common ratio is key to determining whether a geometric series converges.

For the series \(\sum_{n=0}^{\infty} e^{-2n}\), the common ratio is \(e^{-2}\). Calculating gives us approximately 0.135, reflecting a number significantly less than 1, ensuring the series' convergence. This small fraction results in each term being smaller than the last, ensuring they sum towards a limit. Identifying and utilizing the common ratio allows mathematicians to efficiently assess and solve problems related to infinite series.

Sum of Geometric Series

After determining that a geometric series converges, we can find its sum using a simple formula: \(S = \frac{a}{1-r}\). Here, \(a\) is the first term of the series, and \(r\) is the common ratio. This formula arises because each term contributes less and less to the sum.

In our example series \(\sum_{n=0}^{\infty} e^{-2n}\), we found \(a = 1\) and \(r = e^{-2}\). Plugging these into our formula gives \(S = \frac{1}{1-e^{-2}}\). This final sum represents the limit the series converges to, capturing all infinitely many terms in one tidy number. Calculating with the formula helps quickly determine the exact result without having to manually sum each term.