6.4 Working with Taylor Series

Binomial and Taylor series are powerful tools for representing functions as infinite sums. They allow us to approximate complex expressions and solve tricky problems in calculus and beyond. These series are especially useful for tackling differential equations and evaluating integrals that seem impossible at first glance.

Understanding how to develop and apply these series opens up a world of possibilities in mathematics. From estimating function values to solving differential equations, binomial and Taylor series provide elegant solutions to challenging problems across various fields of study.

Binomial Series and Taylor Series

Terms of binomial series

Top images from around the web for Terms of binomial series

• 

  Binomial theorem - Wikipedia View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Binomial Theorem | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Terms of binomial series

• 

  Binomial theorem - Wikipedia View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Binomial Theorem | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia View original

  Is this image relevant?

• 

  Binomial theorem - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

• Infinite series representation of the binomial expression $(1+x)^n$ valid for any real number $n$ and $|x| < 1$
• General form: $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + ...$
• Each term calculated using the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ represents the factorial of $n$
• $k$-th term of the series given by $\binom{n}{k}x^k$

Taylor series for common functions

• Represent functions as an infinite sum of terms based on the function's derivatives at a single point
• Examples of common Taylor series expansions:
  • $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
  • $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$
  • $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$
  • $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...$ for $|x| < 1$
  • $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ...$ for $|x| < 1$
• These expansions are examples of analytic functions, which can be represented by their Taylor series within their radius of convergence

Techniques for Taylor series development

• To develop a Taylor series for a function $f(x)$ centered at $x=a$:
  1. Find the derivatives of $f(x)$ at $x=a$ up to the desired order
  2. Use the Taylor series expansion formula: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$
• Maclaurin series is a special case of Taylor series centered at $x=0$ obtained by setting $a=0$ in the Taylor series formula
• Error estimation techniques can be used to determine the accuracy of a truncated Taylor series approximation

Taylor series in differential equations

• Approximate solutions to differential equations using Taylor series
• Steps to solve a differential equation using Taylor series:
  1. Assume the solution is in the form of a power series: $y = \sum_{n=0}^{\infty} a_n x^n$
  2. Substitute the power series into the differential equation
  3. Equate coefficients of like powers of $x$ to obtain a system of equations for the coefficients $a_n$
  4. Solve the system of equations to find the values of $a_n$
  5. Substitute the coefficients back into the power series to obtain the approximate solution

Complex integrals via Taylor series

• Approximate complex integrals using Taylor series
• Steps to evaluate a complex integral using Taylor series:
  1. Expand the integrand using its Taylor series representation
  2. Integrate the Taylor series term by term
  3. Determine the interval of convergence for the resulting series
  4. If the interval of convergence includes the limits of integration, the integral can be approximated by the sum of the integrated series terms

Applications and Convergence

Taylor series in differential equations

• Example: Consider the differential equation $y' = y$ with the initial condition $y(0) = 1$
  1. Assume the solution is a power series: $y = \sum_{n=0}^{\infty} a_n x^n$
  2. Differentiate the power series: $y' = \sum_{n=1}^{\infty} na_n x^{n-1}$
  3. Substitute into the differential equation: $\sum_{n=1}^{\infty} na_n x^{n-1} = \sum_{n=0}^{\infty} a_n x^n$
  4. Equate coefficients: $a_1 = a_0$, $2a_2 = a_1$, $3a_3 = a_2$, ...
  5. Solve for coefficients: $a_0 = 1$ (from initial condition), $a_1 = 1$, $a_2 = \frac{1}{2}$, $a_3 = \frac{1}{6}$, ...
  6. The solution is: $y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = e^x$

Complex integrals via Taylor series

• Example: Evaluate $\int_{0}^{1} \frac{dx}{1+x^2}$ using Taylor series
  1. Expand the integrand using its Maclaurin series: $\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + ...$
  2. Integrate term by term: $\int_{0}^{1} (1 - x^2 + x^4 - x^6 + ...) dx = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ... \bigg|_{0}^{1}$
  3. Evaluate the series at the limits of integration: $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ...$
  4. The series converges to $\frac{\pi}{4}$, so the integral can be approximated by this value

Convergence and Analysis

• Convergence criteria: Determine the radius of convergence for Taylor series using ratio test or root test
• Asymptotic analysis: Study the behavior of Taylor series approximations as the number of terms approaches infinity
• Laurent series: Generalization of Taylor series that allows for negative powers of $(x-a)$, useful for analyzing functions with singularities