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.

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Taylor series expansion of $e^{x+y}$ about the point $(0,1)$

My question is: what is the Taylor series expansion of $e^{x+y}$ about the point $(0,1)$? I think the standard $e^{x+y} = 1 + x+y + 1/2(x+y)^2$ ... doesn't apply here. Thanks in advance
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2answers
56 views

Upper Bound for $|f^{n}(0)|$ given that $f$ is Analytic

Let $f(x)$ be an analytic function in some neighborhood of $x=0$. $f$ being analytic implies that its has a convergent Taylor series expansion about $x=0$. That is, there exists $R>0$ (radius of ...
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7answers
211 views

Taylor series for $\sqrt{x}$?

I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particual reason no one shows Taylor series for exactly ...
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2answers
33 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
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1answer
59 views

Taylor series of a taylor series: why can we do this?

Suppose we have two functions with the following Taylor series: $$sin(x) = x-x^3/3!+x^5/5!+O(x^6)$$ $$e^x = 1+x+x^2/2+x^3/3!+x^4/4!+x^5/5!+O(x^6)$$ I know, by intuition and because that's what we got ...
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1answer
47 views

Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$

Find the residue at $z=-2$ for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty} (-1)^k\zeta(k+1)z^k$$ Let $z \to -1 - z$ to get: $$\psi(-z) = ...
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Find the McLaurin series for $f(x)=x/(x+1)$

Can you help me find the McLaurin of $f(x)=x/(x+1)$ ? I am new to this mathematical chapter and already tried but I do not think my result is correct.
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1answer
42 views

Taylor expansion about a point

I need help with the following calculus problem: Use completing the square and the geometric series to get the Taylor expansion about ${x=2}$ of ${\frac{1}{x^{2}+4x+3}}$ So far I have the following: ...
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2answers
37 views

Reverse engineering a Taylor expansion

We have the sum: $$S(x) = \frac{x^4}{3(0!)} + \frac{x^5}{4(1!)} + \text{ }...$$ And we are told to sum the series to obtain a finite expression. My guess was to reverse engineer the expression in ...
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the Zassenhaus /Baker–Campbell–Hausdorff formula for cosine.

This question concerns the expansion of non-commutative algebra $[X,Y] \neq 0$ for two operators $X,Y$. One can think of $X$ and $Y$ as some matrices. If $[X,Y] = 0$, we have $$e^{t(X+Y)}= e^{tX}~ ...
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1answer
40 views

Taylor series expansion in calculus of variations

I am reading a book on calculus of variations, so I stumbled upon this integral, which the author expands by taylor series expansion, where $y$ and $y'$ are functions of $x$ and $\tilde{y}(x) = y(x) ...
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0answers
20 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...
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0answers
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Prove that $f$ has a global maximum given a product of Taylor polinomials

Let $f:\mathbb{R}\to\mathbb{R}:f\in C^{\infty},f\left(1\right)=0$ and the product of its Taylor polinomials of order 2 in $x_{0}=0$ and $x_{0}=1$ be $P(x)=x^{4}-2x^{3}+2x^{2}-x$. Prove that $f$ has a ...
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1answer
26 views

Taylor expansion of $f(z)=\frac{z-1}{z^2-3z+3}$

We are given the function $f: \mathbb C \to \mathbb C$ defined by $f(z)=\frac{z-1}{z^2-3z+3}$ Is it possible to define $f$ as its taylor expansion near the point $z=i\sqrt 3$? If so, what is the ...
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1answer
46 views

Finding radius of convergence for taylor series of $\frac{z}{e^z-1}$

Im trying to find the radius of convergence for the taylor sum of the function $\frac{z}{e^z-1}$ around z=0. So far I've found the coffiecients $a_0=1$, $a_1=-\frac{1}{2}$, $a_2=\frac{1}{12}$ and ...
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0answers
12 views

Vanishing terms in the limit of Taylor expansion

$x^*, p \in \mathbb{R}^n$, $f$ convex. $f(x^* + \alpha p) = f(x^*) + \alpha \nabla f(x^*)^tp + o(\alpha)$ Assume $\alpha \nabla f(x^*)^t p < 0$. If we let $\alpha$ go to $0$, we can conclude that ...
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0answers
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Unclear of details of limit for a Taylor expansion of moment generating function - test this Wed!

I keep coming across these limits - the context is moment generating functions and the Central Limit Theorem, but I'm guessing it's a more general question - here is one example (from the proof for ...
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1answer
27 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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1answer
28 views

Determining if a function is real anaytic at the point $a$?

Is there a method, other then using refer to the Taylor series to determine if a real function is analytic at the point $a$. If so please, if possible, could you give a source.
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39 views

from Taylor Series to function

I can't find the logic , how can I find the $f(x)$ and for which x it is defined when I'm given this : $$\sum_{n=0}^\infty(x-1)^n/2^n$$ I see that $x_0=1$ and somehow reminds me of $1/(1-x)$ but I ...
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1answer
67 views

Calculate $\cos(\pi/20)$ with error less then $10^{-5}$

I used MacLaurin series up to n=4, $\lvert R_{4}(x) \rvert < 10^{-5}$ and after I calculated the series with $\frac \pi {20} $ I got something like 0.9876.. and then I checked on the calculator to ...
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3answers
29 views

Shortcut with x0 for taylor expansion

I am confused about the point $x_0$ in Taylor Series Expansion. $f(x) = 1/(1-x)$ and $\sum_n^\infty x^n$ at $x_0=0$ so I thought that if $x_0$=2 I don't need to go through all the process solving ...
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3answers
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If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero?

$$ f: \Re \longrightarrow \Re \ \in C^{\infty} \\ \exists \ L>0: \ \forall x \in \Re, \forall n\in N \\ |f^{(n)} (x)| \le L \\ f(\frac{1}{n})=0 \ \forall n\in N \\ f(x) \equiv 0 $$ Good morning, ...
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2answers
48 views

Convergence of the Power Series of $\log(1+\sin x)$

To find the first few terms in the power series of $\log(1+\sin x)$. The problem went onto expand as $\log(1+x)$ and then further expand by using the $\sin x$ series to get the answer. My doubt is ...
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1answer
53 views

$\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(\nu(x))}{g'(\nu(x))} $ ,the value of the limit: $\lim_{x \to 0^+} \frac{\nu(x)}{x} $

Good evening, I thought a lot about this issue. I think I have to apply Lagrange, Taylor. Can someone help me to calculate this limit? $$f,g \in C^2 [0,1]: \\ f'(0)g''(0) \ne f''(0) g'(0) \\ ...
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1answer
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Can a Taylor Series converge at a point when the remainder term cannot be bounded?

Taylor's Theorem (in the version I'm most familiar with) states that for a function $f$ which is $n+1$ times continuously differentiable on an open neighborhood containing the segment $L$ between $a$ ...
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1answer
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Expand and hence find (Series)

After trying some more questions on Series I'm coming across problems that are rather similar but can't quite grasp what the question is asking for. The question is as follows: Write the first ...
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0answers
27 views

Operator of Taylor series as a distribution?

I would like to prove a statement: $$T:=\sum_{k=0}^\infty a_k\partial_x^k\delta_0\not\in\mathscr{S}'(\mathbb{R}^n)$$ and, in contrast, $$T_n=\sum_{k=0}^n ...
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3answers
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Maclaurin series of $\arctan(x)$ up to degree $4$ [duplicate]

How can I find the Maclaurin series up to degree 4 for: $$\arctan(x)$$ Calculating the derivatives becomes complex very quickly. Is there a special expansion for $\arctan$ like there is for ...
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1answer
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Finding Maclaurin series grade 4

I've tried finding the Maclaurin series grade 4 of the function: $cos(x^2)$ I calculated the four derivatives of the function manually and failed somewhere along the way. Is there an easier way to ...
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1answer
49 views

Find the radius of convergence of the Taylor series of $f(x) = \frac{x-3}{x+2}\ln(5+x)$ at $x=0$

I need find radius of convergence for Taylor series in $x = 0$ (over $\mathbb{R}$) and find $x$'s at which series converges to $f$ $$f(x) = \frac{x-3}{x+2}\ln(5+x)$$ My solution $\ln(5+x) = ...
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218 views

Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

For $x\in\mathbb{R}$, define $\sin (x) = x - x^3/3!+x^5/5!-\cdots$ and $\pi = 4(1-\frac{1}{3}+\frac{1}{5} -\frac{1}{7}+\cdots)$. Then show that $\sin(\pi/2) = 1$ In the prologue of Real and Complex ...
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1answer
92 views

proof for zero function

I am given the following: Let $f$ be a real function, which itself and all its derivatives at $0$ are $0$. Assume there exists $b>0$ such that for all real $x$ and all natural $n$: $|f^{n}(x)|\leq ...
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Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
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2answers
43 views

Generalization of linear approximation? [closed]

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
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71 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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McLaurin series expansion to evaluate a function

I have a maths assignment due for college based on the McLaurin series and don't understand how to do it. I need to use a McLaurin series expansion to evaluate a function. The function is the ...
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44 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
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Remainder of $\ln x$ converges to $0$

I'm learning about power series and struggling to prove If $f(x)=\ln x$ prove that $R_n(f,c)(x)$ converges to $0$ where $c=1$. By some calculating I know that ...
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0answers
17 views

Estimating the error in Taylor series and my attempt

Question is to find number of non zero terms of approximation g (x) which is mclaurin expansion of sin2x .Error is atmost 125\3000 .And my attempt showed answer 3 but textbook says it is 4 .Kindly ...
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1answer
34 views

Taylor series error estimation question

Question is that Taylor series of cosx is restricted to only first two terms and permissible error is 0.54 × 10^(-2) then x can atmost be A) 0.6 B) 0.5 C) 0.4 D) 0.3 My atempt is as follows we need ...
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Modified Bessel function of second kind for imaginary order for small argument

I would like to find an expression for the first terms in the expansion of the modified Bessel function of the second kind $K_n(z)$ for $n\in i\mathbb{R}$ and $z\in \mathbb{R}$ with $z\to 0^\pm$ (I am ...
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1answer
45 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
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2answers
38 views

Mclaurins with $e^{\sin(x)}$

To evaluate $e^{\sin(x)}$ I use the standard series $e^t$ and $\sin(t)$, combining them gives me: $e^t = 1+t+\dfrac{t^2}{2!}+\dfrac{t^3}{3!}+\dfrac{t^4}{4!}+O(t^5)$ $\sin(t) = ...
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3answers
63 views

Limits using Maclaurins expansion for $\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$

$$\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$$ Using Maclaurin's expansion for the numerator gives: $$\left(1+x^2\cdots\right)-\left(x^2-\frac{x^4}{2}\cdots\right)-1$$ And ...
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2answers
54 views

Taylor Expansion of $x\sqrt{x}$ at x=9

How can I go about solving the Taylor expansion of $x\sqrt{x}$ at x=9? I solved the derivative down to the 5th derivative and then tried subbing in the 9 value for a using this equation ...
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2answers
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Proof $|\sin(x) - x| \le \frac{1}{3.2}|x|^3$

So, by Taylor polynomial centered at $0$ we have: $$\sin(x) = x-\frac{x^3}{3!}+\sin^4(x_o)\frac{x^4}{4!}$$ Where $\sin^4(x_0) = \sin(x_o)$ is the fourth derivative of sine in a point $x_0\in [0,x]$. ...
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1answer
17 views

Bounding taylor error

I calculated the polynomial or order $2$ for $\ln(x)$, centered at $x_o=1$, which is: $$\ln(1.3) = \ln(1.0) + \ln'(1.0)(x-1) + \ln''(1.0)(x-1)^2$$ Where the lagrangian error is: $$E(x) = ...
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2answers
52 views

Taylor approximation for $\ln(1.3)$

I have to calculate an approximation for $\ln(1.3)$ using degree $2$ expansion for Taylor polynomial: $$P_2(x) = f(x_0) + f'(x_0)(x-x_0) + f''(x_0)(x-x_0)^2$$ So I can take $x_0 = 1$ and $x = 1.3$ ...
4
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0answers
58 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...