08_Ex 2.8

Exercise Questions

Questions	Question Links
Q1. Apply the Maclaurin series expansion to prove that..	2.8 Q-1
Q2. Show that…	2.8 Q-2
Q3. Show that…	2.8 Q-3

---

Overview

This exercise focuses on applying the Maclaurin series expansion and Taylor series to derive and prove key mathematical functions, including logarithmic, trigonometric, exponential, and square root functions. These expansions provide approximations that are crucial in fields such as calculus, physics, and engineering, enabling solutions to complex problems where exact values are impractical.

---

Key Concepts

• Taylor Series $f(x + h) = f(x) + f'(x)h + \frac{f''(x)}{2!}h^2 + \frac{f^{(3)}(x)}{3!}h^3 + \dots$

• Maclaurin Series $f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots$

• Derivative Calculation:
  Derivatives of functions are calculated step-by-step to determine terms in the series. These include logarithmic, trigonometric, and exponential functions.

• Function Behavior at $x = 0$:
  The series expansion relies on evaluating the function and its derivatives at $x = 0$, providing simplified coefficients.

---

Important Formulas

1. Logarithmic Series:
   $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$

2. Cosine Series:
   $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$

3. Square Root Series:
   $\sqrt{1-x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} + \dots$

4. Exponential Series:
   $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
   $e^{2x} = 1 + 2x + \frac{4x^2}{2!} + \frac{8x^3}{3!} + \dots$

---

Tips and Tricks

• Pattern Recognition:
  Look for alternating signs ($+, -$) and factorials in series terms to simplify expansions.

• Derivative Computation:
  Systematically compute derivatives and evaluate them at $x = 0$. Errors often arise in higher-order terms.

• Shortcut for Exponentials:
  Exponential series have constant derivatives, making their expansions straightforward.

• Common Mistakes:

  • Incorrect evaluation of derivatives at $x = 0$.
  • Misplacing negative signs in logarithmic and square root expansions.

---

Summary

This exercise demonstrates how Maclaurin series provides a framework to approximate functions using polynomial terms. By calculating derivatives and substituting $x = 0$, the expansions are derived, offering practical applications in approximation and analysis.

---

Reference

By Great Science Academy:

---