Questions tagged [taylor-expansion]

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

The Taylor expansion is the power series expansion of a function at a point. It represents a function as an infinite sum with terms calculated from the function's derivatives at that point. More precisely, It is defined as $$ \sum^{\infty}_{n=0}\frac{f^{(n)}(a)}{n!}(x-a)^n=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots $$

It happens often in applications that the Taylor expansion of $f$ at $a$ converges to $f$ (pointwise and locally uniformly) on some neighborhood of $a$: when this happens, the function is said to be analytic at $a$.

Applications:

A Taylor series is an idea used in computer science, calculus, and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. Taylor Series are also used in power flow analysis of electrical power systems (Newton-Raphson method). Multivariate Taylor series is used in different optimization techniques; that is, you approximate your function as a series of linear or quadratic forms, and then successively iterate on them to find the optimal value.

References:

https://en.wikipedia.org/wiki/Taylor_series

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Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the step is derived from second order Taylor…
Libor
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Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values of the constants: $C_0,C_1,C_2,\dots , C_n$. To do…
mauna
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Stuck with Taylor expansion of $f(x+x')$

I know that the Taylor series of $f(x)$ around $a$ is given by: $$f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^2}{2}+\dots=\sum_{n=0}^\infty \frac{f^{(n)}(a) }{n!} (x-a)^n$$ In my textbook I see the following formula for $f(x+x')$ which I however don't…
user104662
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how is the gradient derived here?

I'm taking an online machine learning class and in lecture 9 which covers gradient descent, I can't quite follow how he derives the direction vector of the descent (around the 57:15 mark). He's explaining how for gradient descent, we basically move…
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The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^n \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n
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Finding the Taylor Series of this function

I am trying to find a series expansion of the following function: $$\left(\frac{\log x}{x}\right)^n$$ I need hints or methods for going about doing this. Is it even possible? I am on to something with the general case without the exponent (With…
Ali Caglayan
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Explaining and using the $N$-term Taylor series for $\sin x$

So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer: Explain why the Taylor series containing $N$ terms is: $$\sin x = \sum_{k=0}^{N-1}…
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maclaurin polynomial upper limit

I have the following integral: $$\int_{0}^{1/2} e^{x^2}dx$$ i have approximated the 5th degree maclaurin polynomial of the integral to be: $1+x^2+(1/2)x^4$. I need to obtain an upper bound on the error in the integrand for x in the range $0\le…
user90950
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Taylor's Theorem Problem

This is from my engineering mathematics textbook. Is this version of taylor's theorem correct ? Successive Differentiation, Maclaurin's and Taylor's Expansion of Function $-147$ TAYLOR'S THEOREM Let $f(x)$ be a function of $x$ and $h$ be small.…
Isomorphic
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Natural Logarithm Taylor Series Expansion

f(x)=x$^3$ln(1+2x) Write the first four non-zero terms of the Taylor Series for the above function with x centered at a=0. Using this model: ln(1+x) = Σ$\frac{(-1)^{k}(x)^{k}}{k}$ I get the following... x$^{3}$ln(1+2x) =>…
Jake
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Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just wondering is there a trick for this one or do I…
jjepsuomi
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Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z…
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Rotated functions and taylor series

If one rotates a function such as the sine function about the origin, is there a general method to find the taylor series for the rotated function? Assuming of course that the rotated function is still a single valued function.
PMay
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Proofs using Taylor Series Expansion

I would just like some help on the theory of maths. If a question asks for proof of a function using the Taylor Series Expansion, can you use the Maclaurin Series? Is using the Maclaurin Series regarded as using the Taylor Series Expansion or not?…
Polly
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Need help in Taylor series expansion

In this question, I have to write Taylor's series expansion of the function $f(x) = ln(x+n)$ about x = 0, where n ≠ 0 is a known constant. I have done the following: But my professor handed me back the answer as shown in red above. Could somebody…
Karl
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