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 serise …

I think it is interesting, if we have the formula $\frac{n^m - (n - 1)^ m }{m}$ ~ ${n ^ {m-1}}$ for example $\frac{100^3 - (100-1) ^ 3 }{3} $ = $9900$ ~ ${100 ^ 2}$, if the difference (I'll ...
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Taylor Expansion of Inverse of Difference of Vectors

I am trying to derive the multipole moment of a gravitational potential, but I'm getting stuck on some math I believe. So basically the problem is finding the Taylor Expansion for ...
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33 views

Taylor series of mandelbrot bulb boundaries

What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which ...
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2answers
27 views

Taylor series Expansion

I'm a little confused as to what they are asking. all the examples of taylor series expansion I have seen use x instead and I'm not sure how I would go expanding these series.
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Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
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42 views

The Taylor coefficients of a function of the form $\exp\circ f$, where $f$ is a power series

Let $(a_1, a_2, \dots) \in \mathbb{R}^\infty$ be a fixed sequence of real constants, and suppose the rule $$ x \mapsto \sum_{n = 1}^\infty a_n x^n $$ defines a function from the nonempty open interval ...
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22 views

General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$

What is the general form for the series coefficients of Taylor expansion of $(x+1)^{1/x}$? The first few terms are as follows: $$e-\frac{e x}{2}+\frac{11 e x^2}{24}-\frac{7 e x^3}{16}+\frac{2447 e ...
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29 views

Taylor Series for $\frac{1}{1+e^z}$ and radius of convergence

I have done some manipulation and got that $$\frac{1}{1+e^z} = \sum_{n=0}^\infty \frac{n!}{n!+z^n}$$ by the fact that: $$\frac{1}{1+e^z}= ...
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11 views

Effective ways to calculate multivariable taylor expansion

I need to calculate first 20 members of taylor series for $e^{x^7+y^{11} \cos{(x^{10}+y^8})}$. Are there any ways except the terrible direct way.
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Differentiate a Differential equation

Given the Differential equation $y'=-2xy^{2}$. Find the derivative $\frac{d(y')}{dx}$! My approach, which is not correct according to Wolfram Alpha: Plugging in: ...
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449 views

Exponential function-like Taylor series: what is it?

I have a series $$1+ x+\frac{x^2}{2}+\frac{x^3}{4}+\frac{x^4}{8}...=1+\sum_{n=1}^\infty \frac{x^n}{2^{n-1}}$$ that looks an awful lot like a Taylor series of some kind. If the denominator of the ...
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10 views

Taylor expanion of exponential matrix

I've been reading about Lie groups, and came across the following expansion that left me confused: Let $$ A = e^{i\lambda X_a} \text{ and } B = e^{i\lambda X_b} $$ for matrices $X_a$ and $X_b$, and ...
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3answers
27 views

Errors and Taylor Polynomials

For $g(x)=x^{1/3}$, $a=1$, degree $3$ I found the Taylor polynomial: $$p_3(x) = 1 + (x-1)/3 - ((x-1)^2)/9 + (5(x-1)^3)/81$$ How do I use the error formula for the Taylor polynomial of degree 3 to ...
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37 views

Difficult limit question on Taylor expansion

Let $f:(-1,1) \rightarrow \mathbb{R}$ be a $C^{n+1}$ function. $f$ can be written using Taylor formula with Lagrange remainder as: $$ f(x) = \sum_{k=0}^{n-1} \frac{f^{k}(0)}{k!} x^{k} + ...
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1answer
40 views

Ring of Infinitely Differentiable Functions

Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$. I have managed to prove that this ring is a Principal Ideal Domain. I must also prove ...
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33 views

Show by comparison with an appropriate geometric series that, $e^x-1<\left(\frac{2x}{2-x}\right)$ for 0<x<2.

Show by comparison with an appropriate geometric series that, $e^x-1<\left(\dfrac{2x}{2-x}\right)$ for $0\lt x\lt2$. Can anyone help me with this?
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1answer
11 views

Evaluating irrational values of functions with Taylor series

Calculate the following using Taylor expansion such that the error will be smaller than $10^{-3}$. $\tan 46^\circ$ $(31)^{1/5}$ My problem is that I don't know if I can avoid to use ...
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1answer
43 views

Approximating $e^{\frac 1 {10}}$ with Taylor expansion

Approximate $e^{\frac 1 {10}}$ such that the error won't be larger than $10^{-3}$. I tried to use the expansion for $e^x$ but the error is too large even beyond order 4. So I think the only ...
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32 views

Estimating the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$

Estimate the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$ It seems too easy so I just want to make sure: Since $f(x)-p(x)\le R(x)$ and $R_5(x)=\cos (c) \frac {x^5} {5!}$ So ...
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1answer
24 views

Compute the 100th Bernstein polynomial for $e^x$

I need to find $$B_3 e^x = \sum_{k=0}^{100} e^{k/100}\binom{100}{k} x^k (1-x)^{100-k}$$ I can rearrange this to find $$\sum_{k=0}^\infty e^{k/100} \left(\frac{100!}{k!(100-k)!}\right) ...
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38 views

Taylor Series Clarification

For $\sin(x)$, $e^x$, $\cos(x)$... When we are building the $n$-th taylor polynomial, why is it that we always evaluate the functions first $k$ derivatives at $x=0$? In my textbook when they were ...
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100 views

What is the connection between Taylor series and Chebyshev polynomials?

Can somebody help me find some historical references for the connection between Chebyshev polynomials and the Taylor series for sine and cosine functions? We know that Chebyshev polynomials are used ...
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24 views

Simple vs compound interest rates and Taylor expansion

I am having trouble deciphering a portion from my finance text. Let $i = \text{interest rate}$, $n = \text{Some arbitrary time period}$ and $C = \text{Cash invested}$ And also $C(1+i)^n$ ...
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4answers
69 views

Find a power series for this function

$$f'(x) = 2xf(x) + 4x$$ I need to find the power series for $f(x)$, any hints on how this should be approached?
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31 views

Proving substitution rule of taylor series

Given $f, g$ which are both nth differential-able. How do I show that $f(g)$ is also nth differentai-able ? I tried using chain rule to calculate, but it seems like a mess. Then how can I show that ...
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23 views

why does this power series converges to sinh(x)?

given the infinite sum $$\sum_{n=0}^\infty \frac{ x^{2n+1}}{(2n+1)!}$$ of course, by ratio test, it converges for reals. I know that the answer is $\sinh(x)$ and I've seen how this is derived from its ...
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1answer
28 views

Prove that $\cos(x^2)$ is analytic at $x = 0$

I can't figure out how to go about showing any of the properties required for analytic with such a messy derivative. This is for my real analysis class and I just want to see for this example so that ...
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4answers
113 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation ...
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Finding $\sum\frac{1}{2^n(n+1)}$

What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$ I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$. ...
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472 views

Taylor series of ln(1/(1-z)) around 0

One more taylor/maclurian series problem to which I know the answer of, I just have no idea how to get there (This is as a formal power series, so convergence is not an issue) $$\log \left(\frac 1 ...
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1answer
25 views

N'th coefficient of two taylor series

So, I'm taking a course in Analytic Combinatorics, and the author asserts without proof that the n'th coefficient of $z^n$ for the taylor (Around 0) expansion, for nonnegative integer values of r in ...
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Expand a function in Maclaurin's series.

The function is given with: $$\ln(5\cos^{3}(x))$$ Need to be expanded: $$x^{4}$$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the ...
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25 views

Taylor Series for $e^x(x^2 -x + 1)$

Find the Taylor Series for $e^x(x^2 -x + 1)$ about $x=0$. More importantly, find the COEFFICIENT (for nonzero terms) of the taylor series. The answer says: $$e^x(x^2 -x + 1) = 1 + ...
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2answers
44 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
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For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
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Why can you use the Maclaurin Series for certain cases of function not about 0?

Is it possible to use the Maclaurin Series in a problem like this one (AP Calculus BC Question 6 from a few years ago)? Write the first four nonzero terms and the general term of the Taylor ...
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51 views

Taylor series of $x/(x^2-4x+5)$

I'm supposed to find the Taylor series of this function (I can choose to center it at any A I want): $$f(x)= x/(x^2-4x+5)$$ When I derivate, it only gets more and more confusing. How can I make any ...
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42 views

Convenient notation, or something more?

A little while ago I happened across a curious formula that blew my mind (no idea what it's called): $e^{\frac{d}{dx}}f(x)=f(x+1)$ I played around with it a bit and managed to prove it using the ...
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1answer
29 views

Taylor Series to the Power 1/z

I am attempting to find the Taylor Series for $(\frac{\sin{z}}{z})^{\frac{1}{z^2}}$. While I can plug this into Wolfram and use the output, I want to understand how to calculate the Taylor Series ...
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Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
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Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
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A formal solution using Taylor series

Consider the following: $S_r(n)= 1^r+2^r+...+(n-1)^r$ where $S_r(n)$ satisfies: $S_r(n+1)-S_r(n+1)=n^r$ Now, also consider the Taylor series $f(x+h)=f(x)+hf'(x)+(h^2/2!)f'(x)+..$ which can be ...
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Taylor Series with differentiator operator

Hi guys can anyone show me how the Taylor series can be converted from: $$f(x+h)= f(x)+hf'(x)+...$$ to: $f(x+h)=e^{hD}f(x)$, where $D$ is the differentiation operator. How does the differentiator ...
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2answers
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expand function, taylors series, combinatorics, generation functions

I have to expand $f(z)$ into a formal power series $f(z) = \sum\limits_{k=0}^\infty a_kz^k$ (for $z$ close to 0) $f(z)= \frac{z^3}{1-4z+3z^2}$ I know that: $\frac{1}{1-z} = \sum\limits_{k=0}^\infty ...
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12 views

Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla ...
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31 views

Need clarification on a Taylor polynomial question

$$f(x) = 5 \ln(x)-x$$ second Taylor polynomial centered around $b=1$ is $-1 + 4(x-1) - (5/2)(x-1)^2$ let $a$ be a real number : $0 < a < 1$ let $J$ be closed interval $[1-a, 1+a]$ find upper ...
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36 views

Erroneously Finding the Lagrange Error Bound

Consider $f(x) = \sin(5x + \pi/4)$ and let $P(x)$ be the third-degree Taylor polynomial for $f$ about $0$. I am asked to find the Lagrange error bound to show that $|(f(1/10) - P(1/10))| < 1/100$. ...
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1answer
45 views

Complex Taylor Series Circles of Convergence

I am trying to find the Taylor Series and circles of convergence for three different functions. i) $\frac{\sin{z}}{z}$ which I determined the Taylor series to be $\sum_{n=0}^\infty ...
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32 views

What does the h mean within the Taylor expansion of $y(x_0 +h)$ and $y(x_0 -h)$?

I understand that the Taylor series formula is $$\frac{f^n(a)}{n!}(x-a)^n.$$ I also know that the Taylor series expansion of $$y(x_0 +h)=y(x_0) +hy'(x_0)+\frac{h^2}{2!}y''(x_0)+ ...
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1answer
36 views

first order approximation of scalar function of matrix ( Mahalanobis distance)

I have tried to compute the 1st order approximation using Taylor's expansion of the Mahalanobis distance: $f(\mathbf{X})=\mathbf{a^TXa}$, where $\mathbf{a}\in \mathbb{R}^N$. The function maps ...