What is a Taylor series centered at $a$?

A power series with terms $\frac{f^{(n)}(a)}{n!}(x-a)^n$, formed from derivatives of $f$ at $a$.

What is a Taylor polynomial of degree $n$?

The finite Taylor sum through degree $n$, matching the function and its first $n$ derivatives at the center.

What does the radius of convergence describe?

For a power series, it separates the region of absolute convergence from the region of divergence; boundary behavior is checked separately.

What does it mean for a function to be analytic at a point?

The function equals a convergent power series in some neighborhood of that point.

When is a Puiseux series useful?

It represents certain expansions near algebraic branch points using fractional powers.

CalculusLesson 1 of 2 in the course Sequences and series

Taylor Series: Construction, Convergence, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

This lesson introduces Taylor series as a way to express a function using its derivatives at one point. It also explains Taylor polynomials and the special case called a Maclaurin series.

A Taylor series for a function $f$ centered at $a$ is the power series

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

when the series converges. Here, $f^{(n)}(a)$ is the $n$ th derivative evaluated at $a$ , and $n!$ is the factorial of $n$ . The zeroth derivative means the function itself, so $f^{(0)}(a)=f(a)$ .

All flashcards

What is a Taylor series centered at $a$ ?: A power series with terms $\frac{f^{(n)}(a)}{n!}(x-a)^n$ , formed from derivatives of $f$ at $a$ .
What is a Taylor polynomial of degree $n$ ?: The finite Taylor sum through degree $n$ , matching the function and its first $n$ derivatives at the center.
What does the radius of convergence describe?: For a power series, it separates the region of absolute convergence from the region of divergence; boundary behavior is checked separately.
What does it mean for a function to be analytic at a point?: The function equals a convergent power series in some neighborhood of that point.
When is a Puiseux series useful?: It represents certain expansions near algebraic branch points using fractional powers.

Quiz questions

For a power series centered at $a$ with radius of convergence $R$ , what is guaranteed when $|x-a|<R$ ?Answer: The series converges absolutely.
Which statement distinguishes analyticity from infinite differentiability for real functions?Answer: An analytic function equals its convergent Taylor series near the point; a smooth function need not.
Within a power series' region of convergence, what happens to its radius after term-by-term differentiation?Answer: The radius of convergence remains unchanged.
In a multivariable second-order Taylor expansion, what does the Hessian matrix organize?Answer: The second partial derivatives and their contributions to the quadratic terms.
Which type of expansion can include negative powers near a singularity?Answer: A Laurent series.

Based on the Wikipedia article Taylor series, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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