The Taylor series is a powerful mathematical tool used to represent a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point. This series is particularly useful for approximating functions that may not have a simple closed form, or when calculating the function is difficult.

General Form of a Taylor Series:

For a function $f(x)$ that is infinitely differentiable at a point $a$, the Taylor series of $f(x)$ centered at $a$ is given by:

$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots$

Or in summation notation:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$

Where:

• $f^{(n)}(a)$ is the $n$-th derivative of $f(x)$ evaluated at $x = a$,
• $n!$ is the factorial of $n$,
• $(x - a)^n$ is the difference between $x$ and the center point $a$ raised to the power of $n$.

Common Taylor Series Expansions:

1. Exponential Function $e^x$ around $a = 0$ (Maclaurin series): $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

2. Sine Function $\sin(x)$ around $a = 0$ (Maclaurin series): $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

3. Cosine Function $\cos(x)$ around $a = 0$ (Maclaurin series): $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

4. Natural Logarithm $\ln(1+x)$ around $a = 0$ (Maclaurin series): $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ (This is valid for $-1 < x \leq 1$).

How to Use the Taylor Series:

• Approximation: You can approximate functions by truncating the series to a finite number of terms.
• Convergence: A Taylor series converges to the function if the remainder term (the error made by truncating the series) approaches zero as more terms are included. The radius of convergence determines the interval where this is true.
• Polynomial Approximation: For computational purposes, truncating the Taylor series often gives a good polynomial approximation to a function around the center point.

Example: Taylor Series for $e^x$ at $a = 0$:

The function $e^x$ has the property that all of its derivatives are equal to $e^x$. Evaluating these at $a = 0$ gives:

• $f(0) = e^0 = 1$,
• $f'(0) = e^0 = 1$,
• $f''(0) = e^0 = 1$, and so on.

Thus, the Taylor series of $e^x$ at $x = 0$ (Maclaurin series) is:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Tip:

For practical computations, truncating the series to a few terms can provide a good approximation, but make sure to check the error or remainder for better precision when necessary.

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Do you want to explore a specific example or deeper into how to calculate or use Taylor series? Let me know if you have any questions!

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Further Questions:

1. How do we determine the radius of convergence of a Taylor series?
2. What is the difference between a Taylor series and a Maclaurin series?
3. How can we use Taylor series to approximate functions like $\sin(x)$ or $\ln(x)$?
4. How does the remainder term help in understanding the accuracy of a Taylor series approximation?
5. Can Taylor series be used for functions that are not continuous?

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Tip: For more complex functions, try using symbolic tools (like WolframAlpha or Mathematica) to expand Taylor series quickly and explore higher-order terms!