Problem 59 - Essential Calculus Early Transcendentals

Question

Consider the series $$\sum_{n=1}^{\infty} \frac{n}{(n+1) !}$$ (a) Find the partial sums \(s_{1}, s_{1}, s_{3},\) and \(s_{4} .\) Do you recognize the denominators? Use the pattern to guess a formula for \(s_{n} .\) (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.

Step-by-Step Solution

Answer

The series converges to 1 as the sum.

Step 1: Calculate Partial Sums

To find the partial sums, start by calculating the first few terms of the series: - The general term is \( a_n = \frac{n}{(n+1)!} \). - \( a_1 = \frac{1}{2!} = \frac{1}{2} \) - \( a_2 = \frac{2}{3!} = \frac{1}{3} \) - \( a_3 = \frac{3}{4!} = \frac{1}{8} \) - \( a_4 = \frac{4}{5!} = \frac{1}{30} \) Now, calculate the partial sums: - \( s_1 = a_1 = \frac{1}{2} \) - \( s_2 = a_1 + a_2 = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \) - \( s_3 = a_1 + a_2 + a_3 = \frac{5}{6} + \frac{1}{8} = \frac{40}{48} + \frac{6}{48} = \frac{46}{48} = \frac{23}{24} \) - \( s_4 = a_1 + a_2 + a_3 + a_4 = \frac{23}{24} + \frac{1}{30} = \frac{2875}{2880} = \frac{575}{576} \)

Step 2: Recognize the Pattern

Observe the denominators of the partial sums: 2, 6, 24, 576. These numbers are related to factorials. - \( s_1 = \frac{1}{2} = \frac{1}{2!} \) - \( s_2 = \frac{5}{6} = 1 - \frac{1}{3!} \) - \( s_3 = \frac{23}{24} = 1 - \frac{1}{4!} \) - \( s_4 = \frac{575}{576} = 1 - \frac{1}{5!} \) Assume the pattern follows \( s_n = 1 - \frac{1}{(n+1)!} \).

Step 3: Prove by Mathematical Induction

Use induction to prove \( s_n = 1 - \frac{1}{(n+1)!} \). **Base case:** For \( n = 1 \), \( s_1 = \frac{1}{2} = 1 - \frac{1}{2!} \). This holds true. **Inductive step:** Assume \( s_k = 1 - \frac{1}{(k+1)!} \) holds for \( n = k \). Then, for \( n = k+1 \), \( s_{k+1} = s_k + \frac{k+1}{(k+2)!} \). By assumption, \( s_{k+1} = \left(1 - \frac{1}{(k+1)!}\right) + \frac{k+1}{(k+2)!} \) Simplify using common denominators:\( s_{k+1} = 1 - \frac{1}{(k+2)!} \). Thus, the formula is confirmed by induction.

Step 4: Show Convergence and Find the Sum

Given that the partial sum formula is \( s_n = 1 - \frac{1}{(n+1)!} \), observe that as \( n \to \infty \), \( \frac{1}{(n+1)!} \to 0 \). Therefore, \( s_n \to 1 \). Hence, the infinite series converges to 1.

Key Concepts

Partial Sums

A partial sum in an infinite series is simply the sum of a finite number of terms from the series. This helps us explore how the series behaves as we include more and more terms. If the series is \(\sum_{n=1}^{\infty} a_n\), the \(n\)-th partial sum \(s_n\) is \(a_1 + a_2 + \ldots + a_n\). For the series \(\sum_{n=1}^{\infty} \frac{n}{(n+1)!}\), the partial sums are key to understanding convergence.

Let's look at the partial sums calculated:

• \( s_1 = \frac{1}{2} \)
• \( s_2 = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \)
• \( s_3 = \frac{5}{6} + \frac{1}{8} = \frac{23}{24} \)
• \( s_4 = \frac{23}{24} + \frac{1}{30} = \frac{575}{576} \)

We see a pattern forming in the denominators of these sums. Notice how they relate to factorial expressions. When written in terms of factorials, this relationship becomes clear and is crucial in identifying a general formula for the series.

Mathematical Induction

Mathematical induction is a method used to prove that a given statement is true for all natural numbers. It's like climbing a ladder, step by step:

• **Base Case:** First, establish that the statement holds for the initial value, typically \(n=1\).
  For our task, with \( s_1 = \frac{1}{2} \), we verify the statement \( s_n = 1 - \frac{1}{(n+1)!} \) for \( n=1 \). This gives \( \,1 - \frac{1}{2}\)\, which verifies the base case.
• **Inductive Step:** Assume that the statement is valid for \(n=k\). We need to prove it for \(n=k+1\).
  In our situation, if \(s_k = 1 - \frac{1}{(k+1)!}\), then the statement's truth for \(n=k+1\) becomes \(s_{k+1} = s_k + \frac{k+1}{(k+2)!}\). Substitute the previous sum formula, \(s_k\), and simplify to confirm \(s_{k+1} = 1 - \frac{1}{(k+2)!}\).
  Doing so confirms the pattern continues.
  Induction builds the "domino effect" ensuring validity for all natural numbers.

Using induction properly ensures that a pattern or formula guessed from partial sums is actually correct for the entire series.

Convergence Proof

The ultimate goal with any infinite series is often to determine whether it converges to a limit, and what that limit might be. A series converges if the sequence of its partial sums approaches a particular number (the limit).

For the series \( \sum_{n=1}^{\infty} \frac{n}{(n+1)!} \), the derived formula for the partial sum \( s_n = 1 - \frac{1}{(n+1)!} \) provides a powerful insight. As \( n \rightarrow \infty \), the term \( \frac{1}{(n+1)!} \) approaches zero. This leaves \( s_n \rightarrow 1 \).

This shows the series converges to 1. The elegance of convergence proof lies in engaging with the concept of limits and using formulated expressions like partial sums. By demonstrating how series reach their limiting value, we solidify understanding in calculus fundamentals.