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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$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$.

$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$. Why is the restriction $|x|<1$ or $x=1$? I know from Wikipedia that it is because out of this restriction, the function's "Taylor approximation" is not fair. But how…
Silent
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Taylor (Maclaurin) Series remainder for ${\rm sin}\ (x)$

So I just finished doing this problem and I think the solution I got is wrong, it seems a bit too large. According to my calculations, I need 36 terms. I fear I've made a mistake and I would really appreciate it if anyone could confirm or disprove…
Gsp
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Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this problem
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Linearize a simple ODE

This is homework. I have $\displaystyle \qquad S\frac{dh(t)}{dt} + \frac{1}{R}\sqrt{h(t)} = q(t)$ and need to linearize it. Setting all derivatives to zero, I get the steady-staty value of $h - h_0 = q_0^2R^2$. Then, using Taylor series up to…
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Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives $\sin x \geq x - \frac 16 x^3$ which is true…
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f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor series of $f$, centered at $a$. For each $n \in…
akeenlogician
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Proving that 2 functions are equal/not equal

Prove the equality of $f_1$ and $f_2$ given the following conditions: Problem 1 $f_1(x)$ and $f_2(x)$ are functions of finitely summed sine and cosine functions (e.g. $3\cos2x+\sin5x$), any $x\in[-0.00000001,0.00000001]$ satisfies the following…
TheOnly92
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Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ f(x) = \sum_{n=0}^\infty (-1)^n \cdot…
Vazrael
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Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$ I made a table of derivatives and plugged in…
hax0r_n_code
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Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the simple part. I am having trouble understanding what…
user46372819
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Proof that Polynomials Form a Basis

I'm not even sure this is a true statement, but can someone prove that the polynomials for a basis for continuous functions? This seems to be motivation for Taylor series, and several of the eigenbasis in quantum mechanics.
user82004
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How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of convergence $R > 0$. Then the function $f: B_R(z_0)…
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Does |Taylor Series of $f$ - $f$| Converge Monotonically to $0$?

Suppose that $T_n(x)$ be the sum of the first $n$ terms of the Taylor series of $f$ centered at $a$, and $\lim_{n\to \infty} T_n(b)=f(b)$. Is the difference $|T_n(b)-f(b)|$ decrease monotonically? If yes, why? Otherwise, can you give a…
Behzad
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Computation of the remainder term on a Taylor expansion using contour integrals

I am not really used to the methods of complex analysis, I would like to know for basic monotonic functions like exp(x), log(x), sqrt(x), powers (x^n) and trigonometric functions defined on an real interval, is it possible to use explicit contour…
Yoyo
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Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at $\alpha$ is…
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