Mathematics Foundations/9.5 Taylor Series

9.5 Taylor Series Introduction Taylor series are powerful tools in calculus that allow us to represent functions as infinite sums of terms. These series provide a way to approximate complicated functions using polynomials, which are easier to work with in many mathematical contexts. Definition The Taylor series of a function centered at is given by: This can be written more compactly using sigma notation: where represents the nth derivative of evaluated at . When , the series is called a Maclaurin series: Taylor Polynomials The nth-degree Taylor polynomial is the finite sum of the first terms of the Taylor series: Taylor polynomials provide approximations to the original function, with higher-degree polynomials generally yielding better approximations near . Example 1: Taylor Polynomial for Let's find the 4th-degree Taylor polynomial for centered at . We need to find the derivatives of and evaluate them at : The 4th-degree Taylor polynomial is: Common Taylor Series Several important functions have well-known Taylor series expansions: Exponential Function This series converges for all real values of . Sine Function This series converges for all real values of . Cosine Function This series converges for all real values of . Natural Logarithm This series converges for . Binomial Expansion For non-integer , this series converges for . Example 2: Deriving the Taylor Series for Let's derive the Taylor series for centered at . We need to find the derivatives of and evaluate them at : Continuing this pattern, we can see that the derivatives cycle through , , , and . The Taylor series is: Simplifying: Remainder and Error Estimation When approximating a function using a Taylor polynomial, it's important to understand the error involved. Taylor's Remainder Theorem If has continuous derivatives on an interval containing and , then there exists a point between and such that: This is known as the Lagrange form of the remainder. Error Bound If for all in the interval , then the error in using to approximate is bounded by: Example 3: Error Estimation Estimate the error when approximating using the 3rd-degree Taylor polynomial centered at . The 3rd-degree Taylor polynomial for is: For , the 4th derivative is , which has a maximum absolute value of 1. Using the error bound: Therefore, the approximation has an error of at most . Applications of Taylor Series Taylor series have numerous applications in mathematics and other fields: Numerical Approximations Taylor polynomials provide efficient ways to approximate functions, especially when direct computation is difficult. Solving Differential Equations Taylor series can be used to find solutions to differential equations that cannot be solved by elementary methods. Evaluating Limits Indeterminate limits can often be evaluated by replacing functions with their Taylor series. Example 4: Evaluating a Limit Evaluate We can use the Taylor series for : Substituting: Convergence of Taylor Series A Taylor series may not always converge to the original function for all values of . Radius of Convergence The radius of convergence is the radius of the largest disk centered at in which the Taylor series converges. The series converges absolutely for and diverges for . Interval of Convergence The interval of convergence includes all values of for which the series converges. It is determined by finding the radius of convergence and then checking the endpoints. Example 5: Finding the Interval of Convergence Find the interval of convergence for the Taylor series of centered at . The Taylor series is: Using the ratio test: For convergence, we need , which gives . Checking the endpoints: At , the series becomes , which is the negative of the harmonic series and diverges. At , the series becomes , which is the alternating harmonic series and converges. Therefore, the interval of convergence is . Exercises 1. Find the 4th-degree Taylor polynomial for centered at . 2. Derive the Maclaurin series for and verify the first four terms. 3. Use a 3rd-degree Taylor polynomial to approximate . Estimate the error in this approximation. 4. Find the interval of convergence for the Taylor series of centered at . 5. Evaluate using Taylor series. 6. Find the 5th-degree Taylor polynomial for centered at . 7. Determine the radius of convergence for the Taylor series of centered at . 8. Use Taylor series to show that . 9. Find the Maclaurin series for and determine its radius of convergence. 10. Use a 4th-degree Taylor polynomial to approximate by writing it as and using the binomial series.