# Tagged Questions

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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### Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
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### Finding the limit of: $\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$ using taylor polynomials

no solution provided so I was hoping someone would do a quick look over and make sure it looks ok. Finding the limit of: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
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### Linearization of a function at a point

I have this delay differential equation $$\frac{dx}{dt}=a(x(t)-x(t-1))-b |x(t)|x(t)$$ and I have to make a linearization at the point $\left(\bar{x}(t),\bar{x}(t-1)\right)$, but I cannot figure out ...
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### Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
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### How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$

Determinate the best asymptotic expansion for $x \to 0$ for: $$\sqrt{1-2x+x^2+o(x^3)}$$ How should I procede? In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to ...
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### Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
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### Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
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### How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
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### How do I apply a Taylor expansion of this?

Given $$\frac{1}{r}\left(1+\frac{2\epsilon \cos\theta}{r}\right)^{-1/2}$$ I was told by using Taylor expansion I could get $$1-\frac{2\epsilon \cos\theta}{r}$$ with term of order $\epsilon^2$. Can ...
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### How to derive a Taylor series from the ones we know ($\cos x$, $\sin x$, …)

If we know the Taylor expansion for the $\cos(x)$ function around $0$, how can we use it to derive the Taylor expansion of a similar function ($\cos(x+π/4)$) around $0$? I do know how to get the ...
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$$f(x)=\frac{2x+3}{x^2 -4x+5}$$ on $x=2$. My solution: $t=x-2$ => $x=t+2$ , we get: $f(t)=\frac{2t+7}{t^2+1}$ on $t=0$. then: $(2t+7)\sum_{n=0}^{\infty } {(-t^2)^n} = (2t+7)\sum_{n=0}^{\infty }{(... 2answers 47 views ### Show complex equation of closed curve integral I need to show this equation: $$\frac{1}{2ia} \cdot \oint _{\gamma } \frac{e^{iz}}{z-ia}dz = \frac{e^{-a}}{2ia} \cdot \oint _{\gamma } \frac{1}{z-ia}dz$$ I have an hint to using Taylor. I have no ... 4answers 106 views ### Proof of Leibniz$\pi$formula I found the following proof online for Leibniz's formula for$\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute$y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\... 1answer 28 views ### Is it true that (\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw? Let v,w be vector fields and let \Phi_t be the flow generated by v, that is \frac{d}{dt}\big|_0\Phi_t(x)=v(x). The Lie derivative of w in direction of v is usually defined as \mathcal ... 1answer 39 views ### \Pi_{n=0}^\infty (1-a_n)>0 if and only if \sum a_n < \infty. Let a_n be sequence in (0,1). \Pi_{n=0}^\infty (1-a_n)>0 if and only if \sum_{n=0}^\infty a_n < \infty. First I considered \sum log(1-a_n) and tried to find sum inequality. I ... 2answers 49 views ### Finding certain coefficients in Taylor expansion of \log(1 +qx^2 + rx^3) This exercise is part of the STEP 3 paper from 2014. At a certain point in the problem, we 're supposed find a_n for n = {2,5,7,9} where a_n is the coefficient of x^n in the series ... 0answers 30 views ### Integration in an inequality Does integrating on both the sides of inequality with the same upper and lower limits with respect to same variable somehow affect the inequality. I saw an example lets say, Sin x < x ,x>0 ... 2answers 33 views ### How do I find the radius of convergence for \sum_{n=0}^{\infty}\frac{1}{\sqrt{n}}z^n? I'm a little unsure about methods on finding the radius of convergence of a function. It would be great to get some help on how to approach these kinds of problems. 1answer 40 views ### How to find Taylor series when x_0=0 and radius of convergence for \frac{x}{1+x} for f:(-1,\infty) Through the taylor series formula:$$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$I've got that f(x)=x-x^2+x^3-x^3\dots however my teacher claimed ... 3answers 67 views ### Is the following is true? If that so, give me a proof. -log(1-x)=log(1+e^x)?? Is the following is true? If that so, give me a proof.$$-log(1-x)=log(1+e^x)?$$Give me some value where this equality holds. I dont think so it will be same. Because,$$(1-x)^{-1}=1+x+x^2+x^3+\... 1answer 53 views ### Calculus of rank three tensor Let$A(\alpha)$be a matrix that depends to vector parameter$\alpha$. I want to approximate$A(\alpha+\Delta\alpha)$using Taylor expansion. My work: $$A(\alpha+\Delta\alpha) \approx A(\alpha)+\... 1answer 43 views ### How to find taylor polynomial of a function with two variables? Find the second order Taylor expansion about the point (1,-2) of the function f(x,y) = (x^2 + y)e^{xy}. I begin by computing the matrix of partial derivatives of f. Df(x,y)=(2xe^{xy}+e^{xy}y(x^2+... 0answers 13 views ### How do I specify the input to a Volterra series including the kernels? I have a series of dependent and independent variables. I would like to model their relation using a Volterra/Wiener series. How do I specify: h_n a, b Input vector x_n The kernels for each ... 1answer 31 views ### Does there exist a kernel concept for Taylor expansions? In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ... 0answers 25 views ### How to find the asymptotic expansion of \int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx where x \in N(0,1)? I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ... 1answer 42 views ### Multidimensional taylor series sin (x^3y^2) A homework of mine was to compute the Taylor series of f(x,y)=\sin(x^3y^2) around (0,0) to the 25th order. I assumed, as \sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}, that I ... 2answers 43 views ### Why my calculations aren't right? (Maclaurin series) Good evening to everyone! I tried to calculate \cos\left( x- \frac{x^3}{3} + o(x^4)\right) using the MacLaurin series but instead of getting the final result equal to 1 - \frac{x^2}{2}+\frac{3x^4}... 0answers 37 views ### How to define hypergeometric function {}_1 F_1(-n+1;-n+1;z) for n positive integer Consider a truncated Taylor series of the exponential function to approximate e:$$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$I thought of computing this using the hypergeometric finite series _1 F ... 1answer 49 views ### How to do a Taylor expansion of a vector-valued function Let f:\Bbb R^2\to \Bbb R^2 be given by$$f(x,y):= \left(e^x\sin(x+y),e^{y-x}\tanh(y)\right)$$Find the second-order Taylor expansion of f about (x,y)=(0,0). I know how to find the Taylor ... 1answer 29 views ### taylor series without dissipation I need help. Determine the Taylor series about the point (0,-1, 1) of f(x, y, z) = z^3 - 3z^2 + x^2 + 4yx + 2y + 2z + 16. Note. You must not derive. Thx 0answers 22 views ### Taylor series complex variable [closed] Hello people i need two series of: 2z^2 - 3z with center at z=-i-----------the final answer must be something like this: \sum(z+i)^n and the other one is: (z^2 - 1)/(z) with center at z=... 1answer 42 views ### Taylor series for g(z)=\frac{(z^2 - 1)}{(z)} [closed] i need to find the Taylor series for g(z)=\frac{z^2 - 1}{z} centered at z=5, can somebody help me please? Thanks!! EDITED: The answer must be something like this: Summatory of An(z - 5)^n from ... 1answer 52 views ### When is 1-(1-p)^n \sim pn Let 0<p=p(n)<1 with p=o(1). For which p is it true that 1-(1-p)^n \sim pn? With \sim I mean that they are asymptotically the same, so \frac{1-(1-p)^n}{pn}\rightarrow 1, or at least ... 0answers 73 views ### Maclaurin Expansion of \ln(3+x) I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: \ln\left[3\left(1+\dfrac{x}{3}\right)\right] as opposed ... 0answers 27 views ### Taylor expansions for functions of several variables I need help with this question. a) Determine the Taylor expansions at the origin up to the square Terms of f(x, y, z) = \cosh(x) - \sin(yz) - xy(z - 1)^7 and g(x, y) = e^{-y}/(1-x^2). b) ... 0answers 43 views ### “Binomial Expansion” with a Complex Exponent I want to expand,$$ (X + Y)^s $$for X,Y \geq 0 and s \in \mathbb{C} . Into something of the form,$$ \sum_i X^{p_i} Y^{q_i} $$for p_i, q_i \in \mathbb{C} . I am doing this to expand the ... 1answer 122 views ### Prove \sqrt{1+x} can be represented by a power series I need to show that \sqrt{1+x} can be represented as a power series. I need to prove the equality between the function and its Taylor series, not to prove that the Taylor series of the function, \... 1answer 47 views ### Singularities in the complex plane and expansion of Taylor/Laurent Series The function f(z) = \frac{\cosh(z-3i) -1}{(z-3i)^{5}} has one singular point in \mathbb{C}. I understand that the singular point is an isolated singularity at 3i, and I know there are certain ... 2answers 68 views ### Taylor Series as a linear operator T:C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R})? Can the Taylor series be thought of as either a linear operator T: C^{k} (\mathbb{R} , \mathbb{R}) \to C^{\infty} (\mathbb{R} , \mathbb{R}) given by$$ Tf=\sum^{k}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a)^{... 2answers 75 views ### Calculating a limit with series Good evening to everyone. I have a limit that gave me a lot of trouble and I couldn't find a way to solve it. I tried solving it with series but I couldn't arrive at a result. $$\lim _{x\to 0+}\left(\... 3answers 183 views ### Elementary Proof of Ramanujan Master Theorem I was searching for an elementary proof of the Ramanujan Master Theorem and I found a page from Ramanujan's Notebook on wikipedia which contained the proof. I think that it has some gaps, so can ... 0answers 73 views ### Taylor series around x = 0 of f(x) = \int_0^x\frac{dy}{1+y^4} Find the Taylor series around x = 0 of f(x) = \int_0^x\frac{dy}{1+y^4} and its radius of convergence. It seems like one ought to take the Taylor series of the integrand, and then integrate the ... 0answers 31 views ### Consistency in the definition of cross cumulants Suppose that I have an n\times 1 random vector X=(X_1,X_2,\ldots,X_n)'. For \xi=(\xi_1,\ldots,\xi_n)'\in\mathbb{R}^n, we can define the familiar generating functions$$ M_X(\xi)=E\Big[\exp\Big(\... 2answers 51 views ### Intuition for polynomial bases In my linear algebra course I stumbled upon the following observations. We have some function$f: \Bbb{R} \to \Bbb{R}$,$f = f(x)$.$f(x)$may be composed of elementary functions or not, but in ... 3answers 161 views ### Find the Taylor series of$f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ f\ldots \circ f}}$Define$f(x)=ln(1+x)$. Then$f^{\circ 2}(x)=ln(1+ln(1+x))$, and$f^{\circ 3}(x)=ln(1+ln(1+ln(1+x)))$, etc. Find the Taylor series of$f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ ...
I am supposed to use taylor approximations to avoid loss of significance for the following functions: a) $f(x)=\frac{e^x-e^{-x}}{2x}$ b) $f(x)=\frac{log(1-x)+x*e^{x/2}}{x^3}$ and then find \$\lim_{...