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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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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27 views

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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102 views
+50

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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30 views

Generalization of linear approximation? [on hold]

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
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30 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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43 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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4answers
1k views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
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42 views

Series representation of $1/|x-x'|$ using legendre polynomials

Given two vectors $\mathbf x\in\mathbb{R}^3$ and $\mathbf x'\in\mathbb{R}^3$. Assume: $x = |\mathbf x|$ and $x' = |\mathbf x'|$. Prove that: $$ \frac{1}{|\mathbf x - \mathbf x'|} = \frac{1}{\sqrt{x^2 ...
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15 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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41 views

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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1answer
25 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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7 views

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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0answers
75 views
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Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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1answer
42 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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0answers
48 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 ...
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3answers
51 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
36 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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2answers
47 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
43 views

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
16 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
51 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 $x = 1.3$ ...
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1answer
114 views

Is it possible to detect periodicity of an analytic function from its Taylor series coefficients at a point?

Given the Taylor series $\sum a_k (x - x_0)^k$ of an analytic function, it is possible to determine whether the function is periodic more-or-less directly from the coefficients $a_0, a_1, \ldots$ of ...
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1answer
14 views

Taylor expansion of a logaritmic function

A function is given as $ln (y) = ln(\alpha)-\frac{\lambda}{\gamma}ln(\delta L^{-\gamma}+(1-\delta)K^{-\gamma})$ I need to find the second order Taylor $ln(y)$ around $\gamma=0$. How can it be done ...
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2answers
26 views

Finding the Taylor Series Approximation of $\sin(x)$ at $x=\pi/4$

I'm trying to find a Taylor series approximation for $\sin(x)$ at $\pi/4$. Simply going through the derivatives, I get: \begin{align} f(\pi/4) &= \frac{1}{\sqrt{2}}\\ f'(\pi/4) &= ...
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4answers
28 views

Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
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1answer
19 views

Can every power series be representated as a taylor series?

Can every power series be represented as a Taylor series? More concrete: Given an arbitrary power series $\sum_{n=0}^\infty a_n (x-x_0)^n$, is there always a $C^\infty$-function $f$ such that ...
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1answer
30 views

How can I show that $u=e^{\sigma\sqrt{\Delta t}}$ in the binomial option pricing model

Given that $e^{r\Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2\Delta t$ I would like to show that $u=e^{\sigma\sqrt{\Delta t}}$ I know I must somehow use Taylor's approximation $e^x = 1 + x + ...
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11 views

Taylor coefficents of a function

I'm having some troubles for proving: Prove that in the Taylor Polynomial of f(x,y)=sin(xy) centred in (0,0) just the order 4k-2 coefficents are non zero. k={1,2,..n} I don't think induction is a ...
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1answer
30 views

Taylor polynomials of degree n

I have this math question that states: Find the Taylor polynomials of degree $n$ approximating $ln(1+x)$ for $x$ near $0$. The $n$'s are 5, 7, and 9. $f^{(5)}(0)=24$; I got the derivative to ...
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1answer
19 views

Taylor expansion of Airy function

We know that Taylor expansion is : $ f(x_0 + h) = f(x_0) + h f'(x_0) + .. \ $ I wish to expand the Airy function about it's first root , i.e , $Ai (c_1 - \epsilon ) = Ai (c_1) - \epsilon A_i'(c_1) ...
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1answer
58 views

Definition of inverse function

I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative ...
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2answers
31 views

Represent the function $f(x)=x^{0.3}$ as a Taylor series centered at $5$

Represent the function $f(x)=x^{0.3}$ as a power series $\sum_{n=0}^\infty c_n(x-5)^n$ Find the following coefficients: $c_0$, $c_1$, $c_2$, $c_3$ Here are my answers: $c_0= 5^{0.3} $ ...
2
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4answers
263 views

Limit of function using Taylor's Formula

To find: $$\lim_{x \to 0} \left(\frac{ \sin(x)}{x}\right)^\frac{1}{x}$$ by using Taylor's formula. So I used the Taylor's formula for $\sin(x)$ and got:: $\sin(x) = x - \frac{x^3}{6} + O(x^4)$ ...
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1answer
39 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
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2answers
28 views

Taylor Series type inequality

I need to show that for $x>0$, $1+\frac{x}{2} \ge \sqrt{1+x} \ge 1+\frac{x}{2} - \frac{x^2}{8} $. I used the geometric/arithmetic mean inequality to show that $1+\frac{x}{2} \ge \sqrt{1+x}$ is ...
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0answers
40 views

Asymptotic expansion of integral (Laguerre)

Consider $$L_n = \frac{1}{2\pi i } \oint_{C'} \frac{1}{(1-t)^{\alpha+1} t^{n+1}} e^{-\frac{xt}{1-t}} dt\,\,\,\,(1)$$ where $C'$ is an anticlockwise contour around zero. Now set $\alpha = n$ and I want ...
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1answer
51 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...
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4answers
274 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
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1answer
14 views

Series expansion with quadratic

If i define a function $$F(t)=f􏰀(x(t),y(t))$$ with $$ x(t)=x_0+∆x * t + ∆^2x * t^2$$ $$y(t)=y_0+∆y * t + ∆^2y * t^2$$ $∆$ is the slope $dx/dt$ and $∆^2$ is the 2nd derivative $d^2x/dt^2$ of ...
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1answer
22 views

Maclaurin series of $f(x)=x^3\sin 2x$

I need help finding that maclaurin series for following function. $$f(x)= x^3 \sin2x$$ How do you get to the maclaurin series?
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1answer
26 views

Non-vanishing terms of Maclaurin series for $\log(3-\cos x^2)$

I have to find the first two non-vanishing terms in the Maclaurin series of $$g(x) = \log(3 − \cos(x^2))$$ and that prove $x=0$ is a stationary point. What is a quick way of working out the ...
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7 views

Third order piecewise cubic taylor approximation

I'm trying to approximate a function $d = d(v(t)) = d(x(t),y(t))$ by using a quantised integrator, please see QSS3, page 5. Every step, I know either $x, y$ or both. Both are represented by ...
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1answer
32 views

What's the fourth term in the multivariable Taylor expansion?

For a function $f: \Bbb R^n \to R$, the $2$nd order Taylor expansion is: $$f(\mathbf x+\mathbf h) \approx f(\mathbf x)+ Df(\mathbf x) \mathbf h + \frac{1}{2}\mathbf h^T H(f)(\mathbf x) \mathbf h$$ ...
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1answer
34 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
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1answer
32 views

Bounding the error in the finite difference approximation $\frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x)$

A course problem asks me, assuming that $f$ is $C^3$ on $\mathbb{R}$ (and $f'''$ is bounded and continuous on $\mathbb{R}$), to show that $$\left| \frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x) ...
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1answer
248 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
3
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0answers
40 views

Taylor series of $f(x) = \arctan(x)$ converges to $\arctan(x)$

I have to find out the Taylor series of $f(x) = \arctan(x)$ and prove that it converges to $f(x)$ for any $x \in (-1, 1) $. So far I determined the Taylor series to $T_f(x) = \sum ...
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0answers
32 views

Approximate solutions for quintic equation

The other day I asked a question in here about deriving the equations $$u^2\left(\left(1-s_1\right)+3u+3u^2+u^3\right) =\alpha\left(s_0+2s_0u+\left(1+s_0-s_1\right)u^2+2u^3+u^4\right),$$ where ...
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1answer
18 views

Taylor expansion for left&right limit

I must find the left and the right limits of a function. Can I use the Taylor expansion (at $x_0$) in order to evaluate them?
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2answers
50 views

Can I prove $\sum_{n=2}^{\infty }(1-\frac{1}{n!})\ln^n(2)=\frac{2\ln(2)-1}{1-\ln(2)}$ [closed]

Can I prove $$\sum_{n=2}^{\infty }(1-\frac{1}{n!})\ln^n(2)=\frac{2\ln(2)-1}{1-\ln(2)}$$ depending on Taylor series