1) Introduction: Series Representations of Functions 📚
Power series and Taylor series allow us to represent functions as infinite sums of polynomials, providing powerful tools for approximation and analysis. This topic introduces power series, their convergence, and Taylor series (including Maclaurin series) as specific expansions around a point. These are essential in calculus, physics (e.g., solving differential equations), and engineering for modeling complex behaviors.
We’ll cover:
- Power Series: Definition and convergence intervals.
- Taylor and Maclaurin Series: Constructing series from functions.
- Convergence Tests: Determining where series are valid.
- Applications: Approximating functions and solving problems.
Quick Recap: Infinite series convergence prepares us to represent functions as sums.
2) Power Series 🎓
A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) are coefficients and \( c \) is the center.
Definition 25.1: Power Series
A power series centered at \( c \) is \( \sum_{n=0}^{\infty} a_n (x - c)^n \). It converges within a radius \( R \) (interval \( |x - c| < R \)) and diverges for \( |x - c| > R \), with \( R \) found using the ratio test.
Example: \( \sum_{n=0}^{\infty} x^n \).
Example 1: Radius of Convergence
Find the radius of convergence for \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
- Ratio test: \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{n+1}/(n+1)!}{x^n/n!} \right| = \left| \frac{x}{n+1} \right| \).
- \( \lim_{n \to \infty} \frac{|x|}{n+1} = 0 < 1 \) for all \( x \).
- Radius \( R = \infty \).
Answer: \( R = \infty \).
Example 2: Interval of Convergence
Find the interval for \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} x^n \).
- Ratio test: \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(-1)^{n+1}x^{n+1}/(n+1)}{(-1)^n x^n/n} \right| = \frac{|x|}{n+1} \).
- \( \lim_{n \to \infty} \frac{|x|}{n+1} = 0 < 1 \) for all \( x \), but check endpoints.
- At \( x = 1 \): \( \sum \frac{(-1)^n}{n} \) (alternating harmonic, converges).
- At \( x = -1 \): \( \sum \frac{1}{n} \) (diverges).
Answer: \( (-1, 1] \).
3) Taylor and Maclaurin Series 📐
A Taylor series expands a function around a point \( c \), while a Maclaurin series is a Taylor series centered at \( c = 0 \).
Definition 25.2: Taylor Series
The Taylor series of \( f(x) \) around \( c \) is \( \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n \), where \( f^{(n)}(c) \) is the \( n \)-th derivative. Maclaurin series sets \( c = 0 \).
**Steps**: Compute derivatives at \( c \), form the series, and check convergence.
Example 3: Maclaurin Series for \( e^x \)
Find the Maclaurin series for \( f(x) = e^x \).
- \( f(x) = e^x \), \( f^{(n)}(x) = e^x \), \( f^{(n)}(0) = 1 \).
- Series: \( \sum_{n=0}^{\infty} \frac{1}{n!} x^n \).
Answer: \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \).
Example 4: Taylor Series for \( \sin x \)
Find the Taylor series for \( f(x) = \sin x \) around \( c = \frac{\pi}{2} \).
- \( f(x) = \sin x \), \( f' = \cos x \), \( f'' = -\sin x \), \( f''' = -\cos x \), \( f^{(4)} = \sin x \).
- At \( x = \frac{\pi}{2} \): \( f = 1 \), \( f' = 0 \), \( f'' = -1 \), \( f''' = 0 \), \( f^{(4)} = 1 \).
- Series: \( 1 + 0 \cdot (x - \frac{\pi}{2}) + \frac{-1}{2!} (x - \frac{\pi}{2})^2 + 0 \cdot (x - \frac{\pi}{2})^3 + \frac{1}{4!} (x - \frac{\pi}{2})^4 + \cdots \).
- Simplify: \( 1 - \frac{(x - \frac{\pi}{2})^2}{2} + \frac{(x - \frac{\pi}{2})^4}{24} - \cdots \).
Answer: \( 1 - \frac{(x - \frac{\pi}{2})^2}{2} + \frac{(x - \frac{\pi}{2})^4}{24} - \cdots \).
4) Convergence of Taylor Series 🔍
A Taylor series converges to \( f(x) \) within its radius of convergence, determined by the ratio test on the general term.
Definition 25.3: Radius of Convergence
For \( \sum \frac{f^{(n)}(c)}{n!} (x - c)^n \), the radius \( R \) is found via \( \lim_{n \to \infty} \left| \frac{f^{(n+1)}(c)/(n+1)!}{f^{(n)}(c)/n!} \right| \cdot |x - c| < 1 \).
Example 5: Convergence of \( e^x \)
Find the radius of convergence for \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
- \( a_n = \frac{1}{n!} \), \( a_{n+1} = \frac{1}{(n+1)!} \).
- \( \left| \frac{a_{n+1}}{a_n} \right| = \frac{1/(n+1)!}{1/n!} = \frac{1}{n+1} \).
- \( \lim_{n \to \infty} \frac{1}{n+1} = 0 \), so \( R = \infty \).
Answer: \( R = \infty \).
Example 6: Approximation with Series
Approximate \( e^{0.1} \) using the first three terms of its Maclaurin series.
- Series: \( 1 + x + \frac{x^2}{2} + \cdots \).
- For \( x = 0.1 \): \( 1 + 0.1 + \frac{(0.1)^2}{2} = 1 + 0.1 + 0.005 = 1.105 \).
- Actual \( e^{0.1} \approx 1.10517 \), close approximation.
Answer: \( \approx 1.105 \).
5) Applications of Taylor Series 🔍
Taylor series approximate functions, solve differential equations, and compute values (e.g., \( e \), \( \sin x \)).
Definition 25.4: Applications
Taylor series can replace functions in integrals or differential equations, e.g., \( \int e^x \, dx \approx \int (1 + x + \frac{x^2}{2} + \cdots) \, dx \).
Example 7: Integral Approximation
Approximate \( \int_0^1 e^{-x} \, dx \) using the first three terms of its Maclaurin series.
- Series: \( e^{-x} \approx 1 - x + \frac{x^2}{2} \).
- \( \int_0^1 (1 - x + \frac{x^2}{2}) \, dx = [x - \frac{x^2}{2} + \frac{x^3}{6}]_0^1 = 1 - \frac{1}{2} + \frac{1}{6} = \frac{2}{3} \).
- Actual \( \approx 0.632 \), series \( \approx 0.667 \).
Answer: \( \approx 0.667 \).
Example 8: Differential Equation Solution
Solve \( y' = y \), \( y(0) = 1 \) using a series.
- Assume \( y = \sum a_n x^n \), \( y' = \sum n a_n x^{n-1} \).
- \( \sum n a_n x^{n-1} = \sum a_n x^n \).
- Shift index: \( a_1 + 2a_2 x + 3a_3 x^2 + \cdots = a_0 + a_1 x + a_2 x^2 + \cdots \).
- Match: \( a_{n+1} = \frac{a_n}{n+1} \), start with \( a_0 = 1 \).
- Series: \( 1 + x + \frac{x^2}{2} + \cdots \) (matches \( e^x \)).
Answer: \( y = e^x \).
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Find the Maclaurin or Taylor series and its radius of convergence. 📚
\( f(x) = e^{2x} \) (Maclaurin)
\( f(x) = \cos x \) (Maclaurin)
\( f(x) = \frac{1}{1-x} \) (Maclaurin)
\( f(x) = \sin(2x) \) (Maclaurin)
\( f(x) = \ln(1+x) \) (Maclaurin)
\( f(x) = e^{-x} \) (Maclaurin)
\( f(x) = \cos(3x) \) (Maclaurin)
\( f(x) = \frac{1}{1+x} \) (Maclaurin)
\( f(x) = \sin x \) around \( c = \frac{\pi}{2} \) (Taylor)
\( f(x) = \ln x \) around \( c = 1 \) (Taylor)
\( f(x) = \cos x \) around \( c = \pi \) (Taylor)
Challenging Practice Questions 🌟
Instructions: Solve these advanced problems involving series approximations or applications. 🧠
Find the Taylor series for \( f(x) = e^{-x^2} \) around \( x = 0 \) and determine its radius of convergence.
Approximate \( \int_0^{0.5} \sin(x^2) \, dx \) using the first three terms of its Maclaurin series.
Compute the Taylor series for \( f(x) = \frac{1}{\sqrt{1-x}} \) around \( x = 0 \) and find the interval of convergence.
Determine the Maclaurin series for \( f(x) = \tan^{-1}(x) \) and use it to estimate \( \tan^{-1}(0.2) \).
Solve the initial value problem \( y' = x^2 y \), \( y(0) = 1 \) using a power series solution.
7) Summary & Cheat Sheet 📋
7.1) Power Series
\( \sum a_n (x - c)^n \), converges within radius \( R \).
7.2) Taylor Series
\( \sum \frac{f^{(n)}(c)}{n!} (x - c)^n \), Maclaurin when \( c = 0 \).
7.3) Applications
Approximation, integrals, differential equations.
You’ve mastered power and Taylor series! Congratulations on completing Level 4! 🎉