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/Maclaurin Series

Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}$$ In my first attempt I expanded all ...
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74 views

Second partial derivation of vector function and taylor series

I have vector function: $$ f(x,y) = \begin{pmatrix} (R+r\cos(y))\cos(x) \\ (R+r\cos(y))\sin(x) \\ r\sin(y) \end{pmatrix} $$ I have done Jacobian of that function: $$ f'(x,y) = \begin{pmatrix} ...
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108 views

how to find the asymptotic expansion of the following sum:

I need to determine an asymptotic expansion when $q \rightarrow 1$ of the sum $$S(q)=\sum_{n=0}^{\infty} \frac{q^n}{ (q^n + 1)^2 }.$$ Numerical computations suggest that $S(q)\sim\frac{c}{|q-1|}$ ...
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How to know if a MacLaurin/Taylor Series expansion is good?

This question is motivated by this question. So, given $\frac{1}{e^x + 1}$, the 4th order MacLaurin series $1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4$, although correct in terms of the algebra manipulations, is ...
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304 views

Help find the MacLaurin series for $\frac{1}{e^x+1}$

What is the MacLaurin series up to $x^4$ for $\frac{1}{e^x+1}$? My Attempt: $$\begin{align} \frac{1}{e^x+1} &=(1+e^x)^{-1} \\ &\approx 1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4 \\ \end{align} $$ Since ...
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29 views

Maclaurin series and expressing as a ln(argument)

Found this question in my old homework notes that I did not do at the time! I always wondered how I do this... The first part is a explanation. It is kind of long. Sorry! Here is the actual ...
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Whats wrong in following attempt to write Hermite polynomials in terms of hypergeometric function?

Let's have Hermite polynomials: $$ e^{2tx - t^{2}} = \sum_{n = 0}^{\infty}H_{n}(x)\frac{t^{n}}{n!}. \qquad (1) $$ I need to write it in terms of confluent hypergeometric Kummer function for index $n = ...
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Maclaurin series for $ f(x)=a^x$

My friend is having trouble with these two questions on his homework. I want to help him out but I am not 100% sure how to do these. I took Calculus 3 a while back so its all old memory to me! ...
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Convergence of Taylor's series [duplicate]

Let $f$ be a function defined on $(-a,a)$, such than every derivate $f^{(k)}$, $k\geq 0$, is non-negative for every $x\in(-a,a)$. Prove that it's Taylor's series with the center 0 converges to $f$ at ...
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84 views

McLaurin series of complex function

I've got a function $g(z) = \frac {(1-z)(e^z + e^{-z})}{e^z - e^{-z}}$. I have to find coefficients $c_0, c_1, c_2, c_3$ of McLaurin's series of function $g$ (which is $\sum_{n=0}^{\infty} c_n z^n ...
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290 views

Taylor Series Approximation for degree k Taylor polynomial?

Let $T_k(x)$ be the degree $k$ Taylor polynomial of the function $f(x)=\sin(x)$ at $a=0$. Suppose you approximate $f(x)$ by $T_k(x)$. If $|x|\le 1$, how many terms are needed (that is, what is $k$) ...
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The Maclaurin expansion of $\prod_{k=0}^{n-1}\sin\left(\sqrt{\zeta_n^k}x \right)$.

Specifically, I'm interested in the $x^{3n}$ coefficient. The reason for my curiosity is a proof of the closed form of $\sum_{k \ge 1} \frac{1}{k^{2n}}, n \in \mathbb{N}$. Starting with the ...
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723 views

Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
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49 views

Version of Taylor: $F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h \right \rangle.$

My teacher claimed without proof that Taylor's theorem with remainder implied that for a suitable function $F: \mathbb{R}^n \to \mathbb{R}$, $$F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h ...
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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 ...
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103 views

Compute the 10th derivative

$f(x) = (\cos(5x^2) - 1 )/ x^2 $ at $x = 0$ We were given the hint to use the MacLaurin series for f(x). I get how to do it if it was just $\cos(5x^2)$ but what would I do with the other values in ...
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776 views

Find the Maclaurin series of the function f(x) = (7 x^2) sin (2 x)

Find the Maclaurin series of the function $f(x) = (7 x^2) sin (2 x)$ $(f(x) = \sum_{n=0}^{\infty} c_n x^n) $ That is what is given on the question, we have to fill in 5 blanks $c_3$ to $c_7$ The ...
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46 views

using Taylor's formula in a proof

Prove that $1+\frac{1}{n} < e$ for all $n$ in the natural numbers. How does this connect to Taylor's formula? I know that $e^x > 1+x$ for $x>0$, but then where does Taylor's formula come in ...
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Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
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What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
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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 ...
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44 views

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 ...
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50 views

Maclaurin series of: $ f(x) = {x + 5\over1-x^2}$.

I'm trying to get the Maclauren series of: $ f(x) = {x + 5\over1-x^2}$. I am sure there is some trick here, the result according to Mathematica is: $5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$ ...
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442 views

Taylor's theorem for vector valued functions

I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it): ...
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119 views

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 ...
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1answer
15 views

finding sequence for e converging at some speed

I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the ...
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Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
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85 views

Show the Newton method converges to 0 quadratically?

Using taylor series, show that if $x_n$ converges to a root, $f(x_n)$ usually converges to 0 quadratically. I reached a point I think I need to show that $\lim_{x\to \infty} ...
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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 $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
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138 views

Taylor series about different points implies different interval of convergence?

I'm considering the taylor series of functions whose radius of convergence is non-infinite about different points, and I'm not sure if I'm interpreting this correctly. Suppose, for concreteness, you ...
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83 views

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 ...
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204 views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
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295 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
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36 views

Confusion related to Taylor series approximation

I found this Taylor series approximation given by $f(x_{\alpha}) = f(x) + \nabla f(x)'(x_{\alpha}-x) + o(||x_{\alpha}-x||)$. I didn't get how this $o(||x_{\alpha}-x||)$ term came from. Can anyone ...
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159 views

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 $N$ ...
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What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
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138 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
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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 ...
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225 views

What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
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Taylor evaluation in a product solving a limit

I have the following function, which I am supposed to evaluate: $\lim_{x \to 0}{\frac{(e^{-x^2}-1)\sin x}{x \ln (1+x^2)}}$ My though is to replace sin x by its Maclaurin polynomial, as such: ...
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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 ...
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934 views

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 ...
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Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?

Q1: Can we prove that all zeros of cos(x) are real from the following Taylor series expansion of cos(x)? $$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k} $$ The Riemann $\xi(z)$ function is ...
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238 views

How many iterations of Taylor series for n correct decimal digits

I'm using Taylor series to estimate trigonometric functions. So I need to know exactly how many iterations of Taylor series (say for sine) are needed for n decimal digits precision? (I'm writing a ...
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169 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
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1answer
55 views

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 ...
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393 views

Finding Taylor's expansion for $f(x) = \sqrt{1 + x} -\sqrt{ 1 - x}$

I know I have to find the derivatives of $ f(x) $ (i.e. $f'(x)$ ..) but I'm confused about what to do afterwards .
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1k views

Taylor Series of Hyperbolic Cotangent Coth(x)

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, ...
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1answer
222 views

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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2answers
192 views

Is there a closed form expression for the Taylor series of exp((f(z))?

Given a holomorphic function $f(z) = \sum_{k=0}^\infty f_k z^k/k!$, is there a readable formula for the Taylor series of $\exp(f(z))$? Using the chain and product rules, one can obtain $$\partial_z ...