Problem 8 - Thomas Calculus

Question

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=2}^{\infty} \frac{1}{4^{n}}$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{1}{12} \).

Step 1: Identify the General Term

The general term of the series is given by \( \frac{1}{4^n} \). So the sequence of terms in the series starts from \( n = 2 \).

Step 2: Write Out the First Eight Terms

The first eight terms of the series are obtained by substituting \( n = 2 \) through \( n = 9 \) into \( \frac{1}{4^n} \): - When \( n = 2 \), the term is \( \frac{1}{4^2} = \frac{1}{16} \).- When \( n = 3 \), the term is \( \frac{1}{4^3} = \frac{1}{64} \).- When \( n = 4 \), the term is \( \frac{1}{256} \).- When \( n = 5 \), the term is \( \frac{1}{1024} \).- When \( n = 6 \), the term is \( \frac{1}{4096} \).- When \( n = 7 \), the term is \( \frac{1}{16384} \).- When \( n = 8 \), the term is \( \frac{1}{65536} \).- When \( n = 9 \), the term is \( \frac{1}{262144} \).

Step 3: Recognize the Type of Series

Notice that the series \( \sum_{n=2}^{\infty} \frac{1}{4^n} \) is a geometric series. The first term \( a \) of the series is \( \frac{1}{16} \) and the common ratio \( r \) is \( \frac{1}{4} \).

Step 4: Determine If the Series Converges

For a geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). Here, \( |r| = \frac{1}{4} < 1 \), so the series converges.

Step 5: Calculate the Sum of the Series

The sum of an infinite geometric series is given by the formula: \[S = \frac{a}{1-r}\]where \( a \) is the first term and \( r \) is the common ratio. Substituting the values:\[S = \frac{\frac{1}{16}}{1-\frac{1}{4}} = \frac{\frac{1}{16}}{\frac{3}{4}}\]which simplifies to:\[S = \frac{1}{16} \times \frac{4}{3} = \frac{1}{12}\]

Key Concepts

Series Convergence

When discussing geometric series, one important concept is series convergence. Understanding whether a series converges or diverges is crucial for determining if we can calculate its sum. A series converges if the sum of its infinite terms approaches a specific value or result. Conversely, if the sum of the series increases without bound, it is said to diverge.
A geometric series is characterized by a common ratio between consecutive terms. The convergence depends heavily on this common ratio. For a geometric series to converge, the absolute value of the common ratio, \(|r|\), must be less than 1. If this condition is met, the series will converge, meaning we can find its finite sum.
In the example given, the common ratio is \(\frac{1}{4}\), which is less than 1, so the series converges.

Infinite Series

An infinite series is a sum of infinitely many terms. It often arises in mathematical analysis and calculus. Unlike a simple sequence of numbers, an infinite series represents an ongoing addition of terms. In the context of geometric series, we deal with terms that become progressively smaller when \(|r| < 1\).

• Infinite series are written in the form \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) is a typical general term.
• Understanding infinite series is essential for various applications in mathematics, including calculus, physics, and engineering.

In our case, the series \(\sum_{n=2}^{\infty} \frac{1}{4^n}\) helps illustrate how such infinite sums can be computed despite comprising an endless number of terms. The geometric decrease in terms allows for convergence.

Sum of Series

Once a geometric series is confirmed to converge, the next step involves calculating its sum. The infinite sum of a geometric series can be calculated using the formula:
\[S = \frac{a}{1-r}\]
where \(a\) is the first term, and \(r\) is the common ratio.

• This formula is derived from the property of convergence in geometric series.
• The formula simplifies the otherwise complex task of adding an infinite number of terms.

The example series, \(\sum_{n=2}^{\infty} \frac{1}{4^n}\), highlights how this formula simplifies the calculation. By determining \(a = \frac{1}{16}\) and \(r = \frac{1}{4}\), we apply the formula to find the sum, which in this case equals \(\frac{1}{12}\).

Common Ratio

The common ratio is a key feature of geometric series, defining how each term relates to the previous term. In a geometric sequence, each term is \(r\) times the previous term. Understanding the common ratio can clarify the series' behavior as a whole.

• Identifying the common ratio is paramount in determining whether the series converges or diverges.
• It helps in evaluating the series' sum if convergence is established.

In our problem, the common ratio is \(\frac{1}{4}\). Knowing this number allows us to predict that the terms will shrink by a factor of four with every increase in \(n\). This small absolute value (< 1) confirms convergence, enabling us to find a finite sum.