Applications of Maclaurin Series: Examples for ln(1+x), 1/(1-x), arctan x

📚 Quick Study Guide

• 🍎 The Maclaurin series is a Taylor series expansion of a function about 0. It's represented as: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$
• 🔢 Maclaurin Series for $ln(1+x)$: $ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$, valid for $-1 < x \leq 1$.
• ➗ Maclaurin Series for $\frac{1}{1-x}$: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$, valid for $|x| < 1$. This is a geometric series.
• 📐 Maclaurin Series for $arctan(x)$: $arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$, valid for $|x| \leq 1$.
• 💡 These series are useful for approximating function values, especially when direct computation is difficult.
• 📈 The more terms you include in the Maclaurin series, the better the approximation.

🧪 Practice Quiz

1. Question 1: What is the Maclaurin series representation of $ln(1+x)$?
   1. $x + \frac{x^2}{2} + \frac{x^3}{3} + ...$
   2. $x - \frac{x^2}{2} + \frac{x^3}{3} - ...$
   3. $1 + x + x^2 + x^3 + ...$
   4. $x - x^2 + x^3 - x^4 + ...$
2. Question 2: For what values of $x$ is the Maclaurin series of $\frac{1}{1-x}$ valid?
   1. $x > 1$
   2. $x < 1$
   3. $|x| < 1$
   4. $|x| > 1$
3. Question 3: What is the third term in the Maclaurin series for $arctan(x)$?
   1. $\frac{x^2}{2}$
   2. $\frac{x^3}{3}$
   3. $\frac{x^5}{5}$
   4. $-\frac{x^5}{5}$
4. Question 4: Which function's Maclaurin series is a geometric series?
   1. $ln(1+x)$
   2. $arctan(x)$
   3. $\frac{1}{1-x}$
   4. $e^x$
5. Question 5: If you use the first two terms of the Maclaurin series for $ln(1+x)$ to approximate $ln(1.1)$, what is the approximate value?
   1. 0.10
   2. 0.095
   3. 0.105
   4. 0.11
6. Question 6: What is the coefficient of $x^5$ in the Maclaurin series of $arctan(x)$?
   1. 1
   2. $\frac{1}{5}$
   3. $-\frac{1}{5}$
   4. 0
7. Question 7: What is the value of $f(0)$ for the function $f(x) = arctan(x)$ used in Maclaurin series?
   1. 0
   2. 1
   3. $\frac{\pi}{4}$
   4. Undefined