Determine if the following series converges or diverges. Use any method, and give reasons for your answer. (Hint: First show that (1/n!) ≤ (1/n(n-1)) for n ≥ 2.) M8 1 n! n=2 Show that (1/n!) ≤ (1/n(n-1)) for n ≥2. First rewrite the factorial of n. The factorial of n is n! =n(n-1) ▼
Determine if the following series converges or diverges. Use any method, and give reasons for your answer. (Hint: First show that (1/n!) ≤ (1/n(n-1)) for n ≥ 2.) M8 1 n! n=2 Show that (1/n!) ≤ (1/n(n-1)) for n ≥2. First rewrite the factorial of n. The factorial of n is n! =n(n-1) ▼
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Convergence or Divergence of a Series**
Determine if the following series converges or diverges. Use any method, and provide reasons for your answer.
\[
\sum_{n=2}^{\infty} \frac{1}{n!}
\]
**Hint:** First show that \((1/n!) \leq (1/n(n-1))\) for \(n \geq 2\).
---
1. **Inequality Proof**
Show that \(\frac{1}{n!} \leq \frac{1}{n(n-1)}\) for \(n \geq 2\). First, rewrite the factorial of \(n\).
2. **Factorial Definition**
The factorial of \(n\) is defined as \(n! = n(n-1)(\ldots)\).
The diagram includes a dropdown selection for rewriting \(n!\), which contains the following options:
- \(2.\)
- \(!\)
- \(-1\)
- \(3.\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf00542e-b5b8-4896-9f9f-a2b46551c658%2Fa13467c4-3fb4-43d9-bcca-b206501f2ed2%2Frev61i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Convergence or Divergence of a Series**
Determine if the following series converges or diverges. Use any method, and provide reasons for your answer.
\[
\sum_{n=2}^{\infty} \frac{1}{n!}
\]
**Hint:** First show that \((1/n!) \leq (1/n(n-1))\) for \(n \geq 2\).
---
1. **Inequality Proof**
Show that \(\frac{1}{n!} \leq \frac{1}{n(n-1)}\) for \(n \geq 2\). First, rewrite the factorial of \(n\).
2. **Factorial Definition**
The factorial of \(n\) is defined as \(n! = n(n-1)(\ldots)\).
The diagram includes a dropdown selection for rewriting \(n!\), which contains the following options:
- \(2.\)
- \(!\)
- \(-1\)
- \(3.\)
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