Taylor Series

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Taylor series let us approximate complicated functions using polynomials built from derivatives at a single point. This matters because polynomials are easy to evaluate, differentiate, and integrate, so they give us practical ways to estimate functions in physics, engineering, and calculus. Near the chosen expansion point, a Taylor polynomial can match the original function surprisingly well. As more terms are added, the approximation usually becomes more accurate over a wider interval.

The idea is to use information about a function's value, slope, curvature, and higher derivatives at one point to construct a polynomial with the same local behavior. For a function $f(x)$ expanded about $x = a$ , the Taylor series is $f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$ . The $n$ th degree Taylor polynomial keeps only the first $n + 1$ terms of that series. The error depends on how far $x$ is from $a$ and on the size of the neglected higher derivative terms.

Key Facts

Taylor series about $x = a$ : $f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$
Nth degree Taylor polynomial: $T_n(x) = \sum_{k = 0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k$
Maclaurin series is a Taylor series centered at a = 0
For $e^x$ , $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$
For $\sin(x)$ , $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$
Lagrange remainder: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}$ for some $c$ between $x$ and $a$

Vocabulary

Taylor series: An infinite sum of polynomial terms that represents a function near a chosen center point.
Taylor polynomial: A finite truncation of a Taylor series used to approximate a function.
Expansion point: The value $x = a$ around which the polynomial approximation is built.
Maclaurin series: A Taylor series centered specifically at x = 0.
Remainder: The difference between the actual function value and the Taylor polynomial approximation.

Common Mistakes to Avoid

Using derivatives evaluated at $x$ instead of at the center $a$ , which is wrong because Taylor coefficients must come from $f^{(n)}(a)$ , not changing values.
Forgetting the factorial in the denominator, which is wrong because each term must be divided by n! to match the derivative information correctly.
Assuming the approximation is equally good far from the expansion point, which is wrong because Taylor polynomials are usually most accurate near the center.
Mixing up the function and its polynomial approximation, which is wrong because $T_n(x)$ only matches finitely many derivatives and is not automatically equal to $f(x)$ everywhere.

Practice Questions

1 Find the third degree Maclaurin polynomial for $e^x$ and use it to approximate $e^{0.2}$ .
2 Use the Maclaurin series for sin(x) to build the fifth degree polynomial and estimate sin(0.5).
3 A fourth degree Taylor polynomial centered at x = 0 gives a good estimate for a function near x = 0 but a poor estimate at x = 5. Explain why this can happen in terms of distance from the expansion point and neglected higher order terms.

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