Problem 9 - Algebra 2 and Trigonometry

Question

a. Write each series in sigma notation. b. Determine whether each sum increases without limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that limit. \(\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{(n+1) !}+\cdots\)

Step-by-Step Solution

Answer

The series converges and its finite limit is approximately 0.718.

Step 1: Identify the Pattern

The given series is \( \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{(n+1)!} + \cdots \). Each term in the series is of the form \( \frac{1}{k!} \) starting from \( k=2 \).

Step 2: Write in Sigma Notation

The series can be written in sigma notation as \( \sum_{k=2}^{\infty} \frac{1}{k!} \). This notation expresses the series as a sum of terms from \( k=2 \) to infinity.

Step 3: Analyze Convergence or Divergence

The series \( \sum_{k=2}^{\infty} \frac{1}{k!} \) is similar to the exponential series and its behavior can be analyzed by comparing it to the exponential function \( e^x \). The exponential function \( e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} \) converges to the value \( e \) when \( x=1 \).

Step 4: Determine Behavior of the Series

The given series starts from \( k=2 \), which means it is equivalent to \( e - 1 - 1 \). Since \( e \) is approximately 2.718, the series converges to approximately \( 0.718 \). Therefore, the series has a finite limit and approaches this value.

Key Concepts

Sigma Notation

Sigma notation is a compact way to represent the sum of a sequence or series using the Greek letter Sigma (\( \sum \)). It is especially helpful when dealing with infinite series where listing each term is impractical. The basic structure consists of a lower limit of summation, an upper limit usually set to infinity for infinite series, and the general term of the series. For example, the series \( \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{(n+1)!} + \cdots \) can be expressed as \( \sum_{k=2}^{\infty} \frac{1}{k!} \). Here, \( k \) starts at 2 and increases indefinitely.

• Lower Limit: The starting point of your series, \( k = 2 \) in our example.
• Upper Limit: Where your series stops, often \( \infty \) indicating an infinite series.
• General Term: What's being summed, \( \frac{1}{k!} \) represents each term in the series.

Understanding sigma notation is crucial for tackling not just series but any complex summation problems, enabling concise representation and manipulation of mathematical expressions.

Convergence and Divergence

When investigating a series, it's important to determine if it converges or diverges. A series converges if the sum of its terms approaches a finite limit as more terms are added. Conversely, it diverges if the sum grows without bounds or does not settle to a particular value. Let's explore the key points for these concepts:

• Convergence: For a series to converge, the terms must decrease in size and approach a limit. The concept is crucially illustrated in the series \( \sum_{k=2}^{\infty} \frac{1}{k!} \), which resembles the expansion of \( e^x \) and converges to a limit due to factorial growth in the denominator.
• Divergence: In contrast, a series diverges if its terms don't adequately shrink, causing the sum to increase indefinitely. An example is the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \), known for its divergence.

Analyzing whether a series converges or diverges involves methods such as comparison tests or determining the behavior of the terms. In our series \( \sum_{k=2}^{\infty} \frac{1}{k!} \), the close link to the exponential series helps show its convergence, leading to a finite limit approximation of 0.718.

Factorial Notation

Factorial notation is a way to express the product of an integer and all the positive integers below it. Denoted by the symbol \(!\), it simplifies expressions involving multiplication of consecutive numbers. For any positive integer \(n\), \(n!\) is defined as:\[(n! = n \times (n-1) \times (n-2) \times \ldots \times 1)\]

• Basic Examples: For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). Factorials rapidly increase in size.
• Special Case: \(0!\) is defined to be 1. This might seem counterintuitive but is essential for consistent mathematical definitions, especially in combinatorics and series analysis.

Factorials play a pivotal role in series like \( \sum_{k=2}^{\infty} \frac{1}{k!} \) due to their fast-growing nature, which helps the terms of the series diminish quickly, aiding in the convergence of the series. Understanding and using factorials is essential for working with combinatorial processes and series and is deeply enmeshed in many mathematical formulas.