Taylor and Maclaurin series are powerful tools for representing functions as infinite sums. They allow us to approximate complex functions using simpler polynomial expressions, making calculations and analysis easier in many areas of math and science.

These series are essential for understanding function behavior, estimating values, and solving differential equations. By learning about their properties, convergence, and applications, we gain valuable insights into the nature of mathematical functions and their representations.

Taylor and Maclaurin Series

Defining Taylor and Maclaurin Series

• A Taylor series represents an infinite sum of terms expressed using the derivatives of a function at a single point
• The Taylor series for a function $f(x)$ centered at a point $a$ is given by the formula: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$
• A Maclaurin series is a special case of a Taylor series centered at $a=0$
• The Maclaurin series for a function $f(x)$ is given by the formula: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$
• Maclaurin series are useful for approximating functions near $x=0$ (origin)
• Taylor series can be used to represent and approximate a wide range of functions (polynomials, exponential, trigonometric, logarithmic)

Convergence and Radius of Convergence

• The convergence of a Taylor series determines whether the series approximates the function accurately
• The radius of convergence is the range of x-values for which the Taylor series converges to the function
  • Within the radius of convergence, the Taylor series approximates the function well
  • Outside the radius of convergence, the Taylor series may diverge or not accurately represent the function
• The ratio test can be used to determine the radius of convergence for a Taylor series
• If a Taylor series has a finite radius of convergence, it is valid only within that range (interval of convergence)
• Some Taylor series have an infinite radius of convergence, meaning they converge for all x-values (entire domain of the function)

Deriving Taylor Series Representations

Applying the Taylor Series Formula

• To derive the Taylor series for a function $f(x)$ centered at a point $a$, begin by writing out the general form of the Taylor series
• Take successive derivatives of the function $f(x)$ and evaluate each derivative at the point $a$
  • Find $f'(a)$, $f''(a)$, $f'''(a)$, and so on
• Substitute the values of the derivatives at $a$ into the general form of the Taylor series
• Simplify the expression to obtain the Taylor series representation of the function
• Example: Derive the Taylor series for $f(x)=e^x$ centered at $a=0$ (Maclaurin series)
  • $f(0)=e^0=1$, $f'(0)=e^0=1$, $f''(0)=e^0=1$, $f'''(0)=e^0=1$, ...
  • Substituting into the Maclaurin series formula: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$

Manipulating Taylor Series

• Taylor series can be manipulated using standard algebraic operations (addition, subtraction, multiplication, division)
• When adding or subtracting Taylor series, add or subtract the corresponding coefficients of like terms
• When multiplying Taylor series, use the Cauchy product formula to multiply the coefficients
• Dividing Taylor series involves finding the reciprocal series and then multiplying
• Composition of Taylor series can be performed by substituting one series into another
• These manipulations allow for deriving Taylor series of more complex functions from known series (exponential, trigonometric, logarithmic)

Maclaurin Series for Common Functions

Exponential and Logarithmic Functions

• The Maclaurin series for the exponential function $e^x$ is: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
• The Maclaurin series for the natural logarithm function $\ln(1+x)$ is: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$
  • Valid for $-1 < x \leq 1$
• The Maclaurin series for the exponential function $a^x$ (where $a>0$ and $a \neq 1$) is: $a^x = 1 + (\ln a)x + \frac{(\ln a)^2x^2}{2!} + \frac{(\ln a)^3x^3}{3!} + ...$
• The Maclaurin series for the logarithmic function $\log_a(1+x)$ (where $a>0$ and $a \neq 1$) is: $\log_a(1+x) = \frac{1}{\ln a}\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...\right)$
  • Valid for $-1 < x \leq 1$

Trigonometric Functions

• The Maclaurin series for the sine function $\sin(x)$ is: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$
• The Maclaurin series for the cosine function $\cos(x)$ is: $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$
• The Maclaurin series for the tangent function $\tan(x)$ is: $\tan(x) = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + ...$
  • Valid for $-\frac{\pi}{2} < x < \frac{\pi}{2}$
• The Maclaurin series for the hyperbolic sine function $\sinh(x)$ is: $\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + ...$
• The Maclaurin series for the hyperbolic cosine function $\cosh(x)$ is: $\cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + ...$

Function Approximation with Taylor Polynomials

Constructing Taylor Polynomials

• A Taylor polynomial is a finite sum of terms from a Taylor series, used to approximate a function near a given point
• The nth-degree Taylor polynomial for a function $f(x)$ centered at $a$ is denoted by $P_n(x)$ and is given by the formula: $P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n$
• To approximate a function using a Taylor polynomial, choose an appropriate degree $n$ and center point $a$ based on the desired accuracy and region of interest
• Compute the derivatives of the function up to the nth order and evaluate them at the center point $a$
• Substitute the values of the derivatives into the formula for the nth-degree Taylor polynomial
• The resulting polynomial $P_n(x)$ approximates the function $f(x)$ near the point $a$, with accuracy increasing as $n$ increases
• Example: Construct a 3rd-degree Taylor polynomial for $f(x)=\sin(x)$ centered at $a=0$ (Maclaurin polynomial)
  • $P_3(x) = x - \frac{x^3}{3!}$

Error Analysis and Bounds

• The error in approximating a function $f(x)$ by its nth-degree Taylor polynomial $P_n(x)$ is given by the remainder term $R_n(x)$
• The Taylor remainder theorem provides an upper bound for the absolute value of the error: $|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$
  • $M$ is the maximum value of the $(n+1)$th derivative of $f(x)$ on the interval between $a$ and $x$
• Lagrange error bound: If $|f^{(n+1)}(x)| \leq M$ for all $x$ between $a$ and $x_0$, then $|R_n(x_0)| \leq \frac{M}{(n+1)!}|x_0-a|^{n+1}$
• Cauchy error bound: If $|f^{(n+1)}(x)| \leq M$ for all $x$ within a radius of $R$ from $a$, then $|R_n(x)| \leq \frac{M}{(n+1)!}R^{n+1}$
  • Valid for $|x-a| < R$
• These error bounds help determine the accuracy of Taylor polynomial approximations and guide the choice of the degree $n$