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5n (c) Σ(-1)"-1 3n-1 n=1

5n (c) Σ(-1)"-1 3n-1 n=1

Calculus: Early Transcendentals
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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#3 part c

**Exercises and Solutions** **1. Evaluate the Limits:** (a) \(\lim_{n \to \infty} \frac{1}{n}\) (b) \(\lim_{n \to \infty} \frac{n}{n^2 + 1}\) (c) \(\lim_{n \to \infty} \frac{n}{2^n}\) (d) \(\lim_{n \to \infty} n e^{-n^2}\) (e) \(\lim_{n \to \infty} \frac{n}{(\ln n)^2}\) (f) \(\lim_{n \to \infty} \frac{(-1)^n \sqrt{n}}{n + 1}\) (g) \(\lim_{n \to \infty} \left(\sqrt{n^2 + n} - \sqrt{n^2 + 1}\right)\) (h) \(\lim_{n \to \infty} \frac{2^n}{n!}\) **Answers:** (a) \(\frac{1}{2}\) (b) \(0\) (c) \(1\) (d) \(0\) (e) \(\infty\) (f) \(0\) (g) \(\frac{1}{2}\) (h) \(0\) **3. Determine whether the series converges or diverges. If it converges, find its sum.** (a) \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{4^n}\) (b) \(\sum_{n=0}^{\infty} \frac{(-3)^n}{5^{n+1}}\) (c) \(\sum_{n=1}^{\infty} (-1)^{n-1} \frac{5^n}{3^{n-1}}\) (d) \(\sum_{n=1}^{\infty} \frac{2}{4n^2 - 1}\) (e) \(\sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right)\) **Answers:** (a) 12 (b) \(\frac{1}{8}\) (c) divergent (d) 1 (e) \(-\infty\),
Transcribed Image Text:**Exercises and Solutions** **1. Evaluate the Limits:** (a) \(\lim_{n \to \infty} \frac{1}{n}\) (b) \(\lim_{n \to \infty} \frac{n}{n^2 + 1}\) (c) \(\lim_{n \to \infty} \frac{n}{2^n}\) (d) \(\lim_{n \to \infty} n e^{-n^2}\) (e) \(\lim_{n \to \infty} \frac{n}{(\ln n)^2}\) (f) \(\lim_{n \to \infty} \frac{(-1)^n \sqrt{n}}{n + 1}\) (g) \(\lim_{n \to \infty} \left(\sqrt{n^2 + n} - \sqrt{n^2 + 1}\right)\) (h) \(\lim_{n \to \infty} \frac{2^n}{n!}\) **Answers:** (a) \(\frac{1}{2}\) (b) \(0\) (c) \(1\) (d) \(0\) (e) \(\infty\) (f) \(0\) (g) \(\frac{1}{2}\) (h) \(0\) **3. Determine whether the series converges or diverges. If it converges, find its sum.** (a) \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{4^n}\) (b) \(\sum_{n=0}^{\infty} \frac{(-3)^n}{5^{n+1}}\) (c) \(\sum_{n=1}^{\infty} (-1)^{n-1} \frac{5^n}{3^{n-1}}\) (d) \(\sum_{n=1}^{\infty} \frac{2}{4n^2 - 1}\) (e) \(\sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right)\) **Answers:** (a) 12 (b) \(\frac{1}{8}\) (c) divergent (d) 1 (e) \(-\infty\),
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