HomeTextbook AnswersMathCalculusCalculus (3rd Edition)Chapter 11 - Infinite Series - Chapter Review Exercises - Page 593113
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Appendix A Appendix C Infinite Series - 11.1 Sequences - Preliminary Questions Infinite Series - 11.1 Sequences - Exercises Infinite Series - 11.1 Sequences - Exercises Infinite Series - 11.1 Sequences - Exercises Infinite Series - 11.2 Summing and Infinite Series - Preliminary Questions Infinite Series - 11.2 Summing and Infinite Series - Preliminary Questions Infinite Series - 11.2 Summing and Infinite Series - Exercises Infinite Series - 11.2 Summing and Infinite Series - Exercises Infinite Series - 11.2 Summing and Infinite Series - Exercises Infinite Series - 11.2 Summing and Infinite Series - Exercises Infinite Series - 11.3 Convergence of Series with Positive Terms - Preliminary Questions Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises Infinite Series - 11.4 Absolute and Conditional Convergence - Preliminary Questions Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Preliminary Questions Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises Infinite Series - 11.6 Power Series - Preliminary Questions Infinite Series - 11.6 Power Series - Exercises Infinite Series - 11.6 Power Series - Exercises Infinite Series - 11.6 Power Series - Exercises Infinite Series - 11.7 Taylor Series - Preliminary Questions Infinite Series - 11.7 Taylor Series - Exercises Infinite Series - 11.7 Taylor Series - Exercises Infinite Series - 11.7 Taylor Series - Exercises Infinite Series - Chapter Review Exercises Infinite Series - Chapter Review Exercises Infinite Series - Chapter Review Exercises 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
Chapter 11 - Infinite Series - Chapter Review Exercises - Page 593: 113
$$e^{4 x}=\sum_{n=0}^{\infty} \frac{(4 x)^{n}}{n !}=\sum_{n=0}^{\infty} \frac{4^{n}}{n !} x^{n}$$
Given $$ f(x)=e^{4x}$$ first we find for $e^x$ , since \begin{align*} f(x)&=e^x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f(0)=1\\ f'(x)&=e^x\ \ \ \ \ \ \ \ \ \ \ \ \ f'(0)=1\\ f''(x)&=e^x\ \ \ \ \ \ \ \ \ \ \ f''(0)=1\\ f'''(x)&=e^x\ \ \ \ \ \ \ \ \ \ \ \ f'''(0)=1\\ f^{(4)}(x)&=e^x \ \ \ \ \ \ \ \ \ \ \ \ \ f^{(4)}(0)= 1 \end{align*} Then \begin{align*} f(x) &=f(a)+\dfrac{f'(a)}{1!}(x-a)+\dfrac{f''(a)}{2!}(x-a)^2+\dfrac{f'''(a)}{3!}(x-a)^3+.....+\dfrac{f^{(n)}(a)}{n!}(x-a)^n\\ &=f(0)+\dfrac{f'(0)}{1!}(x)+\dfrac{f''(0)}{2!}(x)^2+\dfrac{f'''(0)}{3!}(x)^3+.....+\dfrac{f^{(n)}(0)}{n!}(x)^n\\ &= 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots \frac{x^n}{n!} \\ &=\sum_{n=0}^{\infty}\frac{x^n}{n!} \end{align*} To find the Taylor series for $f(x)=e^{4x}$, replace $x$ by $4x$, we get $$e^{4 x}=\sum_{n=0}^{\infty} \frac{(4 x)^{n}}{n !}=\sum_{n=0}^{\infty} \frac{4^{n}}{n !} x^{n}$$
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Infinite Series - 11.1 Sequences - Preliminary Questions
Infinite Series - 11.1 Sequences - Exercises
Infinite Series - 11.2 Summing and Infinite Series - Preliminary Questions
Infinite Series - 11.2 Summing and Infinite Series - Exercises
Infinite Series - 11.3 Convergence of Series with Positive Terms - Preliminary Questions
Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises
Infinite Series - 11.4 Absolute and Conditional Convergence - Preliminary Questions
Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises
Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Preliminary Questions
Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises
Infinite Series - 11.6 Power Series - Preliminary Questions
Infinite Series - 11.6 Power Series - Exercises
Infinite Series - 11.7 Taylor Series - Preliminary Questions
Infinite Series - 11.7 Taylor Series - Exercises
Infinite Series - Chapter Review Exercises