Questions tagged [taylor-expansion]

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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Exponential equation (volatility matching in binomial tree model of option pricing).

Hull's book $\textit{'Options, Futures and Other Derivatives'}$ says that if we solve this equation $e^{r \Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2 \Delta t$ with respect to $u$ and $d$, we get $...
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Proof of an exponential inequality

This seems to be obvious but I am having a hard time proving it. Any insight greatly appreciated. Statement: Prove for every $b,x \in \mathbb{R}$ such that $b\geq 1$, $|x|\leq b$, it holds that $$(...
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What is the Taylor Expansion of a function [closed]

What is the Taylor expansion of the function $$G(x)=\frac{1}{(1-x^{50})(1-x^{25})(1-x^{10})(1-x^5)(1-x)}\quad ?$$
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What is the radius of convergence for the Taylor expansion of $\frac{e^x\sin(x)}{x^2+25}$ about $x=0$?

I would like to know how I could go about finding the radius of convergence for the Maclaurin series of $\frac{e^x\sin(x)}{x^2+25}$. I am familiar with how to find the radius of convergence of more ...
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Compute $\lim_{(x,y) \to (0,0)} \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2}$

Does the limit $$\lim_{(x,y) \to (0,0)} \left( \frac{x^2 - 2\cos(y) + 2}{y^2 - 2\cos(x) + 2} \right) $$ exist? I think it does and it's equal to $1$, but I don't know how to prove it. I tried to use ...
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How do I approximate $\sum_{i=2}^b \frac{\csc(i x)}{i}$?

If I take the power series around point $b$ of the following sum: $\sum_{i=2}^b \frac{\csc(i x)}{i}$ Series[Sum[Csc[i *x]/i, {i, 2, b}], {x, b, 3}] (...
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Taylor expansion of small parameter

If I have an expression $$\cos^2{(\bar\theta(1+\sigma m(x_3/l))} $$ where $$\bar\theta <<1$$ and $$m(x_3/l)$$ is a N(0,1) random variable, can I expand in small theta to obtain $$\cos^2{(\bar\...
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Order of Taylor polynomial in numerical derivative

I'm stuck on a question regarding numerical methods. The aim of this question is to approximate $ f''(x) + f'(x) $ with the highest possible order of truncation error using $Q(h) = \frac{\alpha_{-1} ...
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Expanding random N(0,1) variable

If I have an expression $$\frac{1}{1+\sigma m(z/l)}$$ where $m(z/l)$ is a random $N(0,1)$ variable, $\sigma$ is dimensionless, can I rewrite this via an expansion to bring up the random variable on ...
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Proving limit of f(x) - Tnf(x) (Taylor) is zero, in multivariable calculus

So as we all probably know, for $ f : \mathbb{R} \to \mathbb{R}$, and $T_nf(x)$ the taylor polynomial at a given point a, we have $\lim\limits_{x \to a} \frac{f(x) - T_nf(x)}{(x-a)^n} = 0$. Our ...
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Stuck trying to find the value of this limit using Taylor series.

So I'm having trouble trying to find the value of the limit $$\lim_{x \to 0}\frac{(\sqrt{2}-\sqrt{1+\cos(x)})(e^x-e^{-x})}{\sin^3(x)}$$ using Taylor series (the problem explicits it). Well, this is ...
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Taylor series and calculating a sum with its help

So this function is given: $$f(x)=xe^{-2x}$$ I have to calculate its Taylor series around $x=1$ and use it in some way to calculate the following sum: $$\sum_{n=1}^{\infty} \frac{n+2}{(2n)!!}$$ I ...
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Extending the Binomial Series of $(a+x)^{k}$ to Cases Where $a<0$ Via Substitution

I was re-visiting the binomial series expansion, and noticed that one author took special pains to restrict the general form of expression like $$ (a+x)^k, $$ saying that the results for $(1+x)^...
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Theorem about local maximum using Taylor polynomial

I have a few questions regarding the theorem attached below. Why does functions $\frac{f(x)}{(x-a)^n}$ and $\frac{f^{(n)}(a)}{n!}$ have an equal sign if $x$ is sufficiently close to $a$? Is it only ...
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Is Convergence of the Maclaurin Series Sufficient to Prove the Convergence of the Taylor Series?

When demonstrating the convergence of the Taylor series of a function like $e^x$, to the function (i.e. $R_n(x) \to 0$) is it sufficient to prove convergence of the Maclaurin series? If the radius of ...
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Taylor's theorem $\lim_{x \to a} \frac{f(x)-P_{n,a}(x)}{(x-a)^n}=0$

First, I wanted to ask whether this theorem concerns the remainder of Polynomial at point $a$ and order n-th? I also wanted to know in general why this is important that the difference between ...
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Taylor polynomials respect derivatives

I want to prove that the derivative of the $n$th order Taylor polynomial is the $n-1$th order Taylor polynomial of the derivative. More specifically: Suppose $f: \mathbb{R} \to \mathbb{R}$ is $n$ ...
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Error estimation for taylor series of $\ln(1+x)$

It is claimed that $$ \ln(1+x) = \sum_{k=0}^{\infty}(-1)^{k-1}\frac{x^k}{k}$$ for all $x \in (-1,1]$. My reference then continues by estimating the error for partial sums of this series: $$f(x)= ...
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Taylor Polynomial of $e^x$

I have attached an image of section in Spivak's "Calculus", where the author explains how the difference between a function an it's Taylor's polynomial at a point $a$ is smaller than difference ...
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Intuitive understanding of Taylor's Inequality (Lagrange's remainder)

I have read through many of the (many) posts related to Taylor's Inequality / Remainder; however, I wondered if there were an intuitive way of understanding the error from point-of-view of series / ...
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“Incomplete” Tricomi Hypergeometric Function

I've come across this integral: \begin{equation} I(x)=\frac{1}{\Gamma(a)}\int_x^{\infty}t^{a-1}(1+t)^{b-a-1}e^{-zt}dt \end{equation} a few times now, but I cant seem to get anywhere with it. I've ...
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Taylor series for $e^{a x} J_0 (b x)$

How to derive the general term for the Taylor series around $0$ for this function? I found, using Wolfram Alpha, that: $$e^{a x} I_0 (|a| x)= \sum_{n=0}^\infty \frac{(2n)!}{n!^3} \frac{(a x)^n}{2^n}$...
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Limit of sequence using taylor's formula on trigonometric functions

I had the following question in an exam. I wasn't able to solve the problem and I don't understand the solution. I am supposed to find the limit of the following sequence: $$ \lim_{x\to 1} \frac{\...
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Finding second-order taylor's series [closed]

Find second-order Taylor’s series approximation of the function $$F(x,y)=e^y(x-1)^2 + x$$ at the points (a) $(0,0)$ and (b) $(1,1)$. I appreciate for any help. I'm not confidence with my solve ...
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Solving limits by Taylor series

Perhaps I didn't fully understand the concept of Taylor series. I would like to compute $$\lim_{x \to 1} \frac{\ln x}{x^{2}-1}$$ using Taylor expansion around the right point (point is not given). ...
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Taylor's theorem and Lagrange remainder.

So, I have a few questions about Taylor's theorem. The first case when $n=0$ is clear to me, but then when the $n= 1$ is considered, they say that $E_1$ is simply the error formula between the ...
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How should I substitute in Taylor series properly?

I am currently studying Taylor series and I can't figure out how to properly compute the Taylor Series of a composition of functions. I have read that the TS of $f(g(x))$ is simply the TS of $f(x)$ ...
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Determine $a, b, c, d$, where $\int_0^x{\frac{t}{\arctan(t)}}dt=a+bx+cx^2+dx^3+o(x^3)$

Let $$f(x)=\int_0^x{\frac{t}{\arctan(t)}}dt=a+bx+cx^2+dx^3+o(x^3)$$ be the function defined by the above integration where $x \not= 0$. The exercise is to determine the constants $a, b, c, d$, but I ...
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Proof that the taylor expansion of a composite function the same as the one acheived by substituition?

I have seen this question Taylor expansion of composite function and it does show that they are equal but the intuition is not provided there. I can't understand why should they be the same. So is ...
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Prove $e^{x \cos(x)}=1+x+\frac{x^2}{2} - \frac{x^3}{3}-\frac{11x^4}{24}- \frac{x^5}{5} + \cdots$

How do we do this question using Maclaurin's Series. I tried expanding it by putting $x \cos x$ in place of $x$ in Maclaurin's expansion of $e^x$, and then using multinomial theorem to open the ...
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Question about zeros of partial sums of Taylor series of Riemann xi-function

Consider the entire function $ \psi(z) $ whose Taylor series about 1/2 "looking like" the Riemann xi-function $ \xi(z)=\sum_{n=0}^{\infty}a_{2n}(z-1/2)^{2n} $, but with different coefficients $a\...
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Using Taylor's theorem to prove Cauchy Riemann equations

My question is about Taylors theorem but in the context of proving the following theorem (C-R equations). Let $f = u + iv$ be defined on domain $D$ in $\mathbb{C}$, where $u$ and $v$are real-valued. ...
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Give a general formula for nth and mth derivative in (0,0) using Taylor.

Given $f(x,y)=\frac{x}{1-2xy^{2}}$, how can I give a general formula for $D_{n,m}=\frac{\partial f^{n+m}}{\partial x^{n}\partial y^{m}}(0,0),\:\:\:n,m\geq0$ using Taylor? Where should I begin?
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Taylor polynomial around $x=0$ for $\ln(2+x)$

For my homework I got this question: explain what is wrong with this statement: if $f(x)=\ln(2+x)$, then the second-degree taylor polynomial approximating $f(x)$ around $x=0$ has a negative constant ...
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Expansion of $\frac{1}{(1 - e^{-x})}$ [duplicate]

As Taylor's series give $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$ Similarly, can someone please help me to find expansion of the term $\frac{1}{(1 - e^{-x})}$? Where $0 < e^{-x} < 1$ Thank ...
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Objective function Taylor series derivative

Deep learning Book, page 228 ...We will further simplify the analysis by making a quadratic approximation to the objective function in the neighborhood of the value of the weights that obtains ...
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Taylor expansion of $\sqrt{n-k}$

I am reading a paper which casually assumes the asymptotic $\sqrt{n-k} \simeq \sqrt{n}-\frac{k}{2\sqrt{n}}$. This expression is what Wolfram calls Taylor expansion at infinity and from what I ...
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Binomial Approximation with Small Exponent

Find an approximation for the expression $ (1 + x)^n $, where $ 0 < x < C $ and $ n $ is positive but small. $ C $ is arbitrarily large (<< 1000) and $ n $ is arbitrarily small ($ n <&...
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multi-step method, consistency order with Taylor expansion

We search the highest order for the multi-step process $y_{k+2}-(1+\alpha)y_{k+1}+\alpha y_k = h(\frac{3-\alpha}{2}f_{k+1}-\frac{1+\alpha}{2}f_k)$. It is $f\in C^3$. We find $\alpha$ with ...
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Taylor expansion of a field in polar coordinates

I'm trying to write the second Taylor expansion of a field defined on a plane using cartesian and polar coordinates. The coordinate system independent equation is: $$f({\bf x})=f({\bf x_o})+({\bf x-...
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How to prove that for all non-negative $\forall x\in\mathbb R: x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}$?

I'm trying to prove that for all non-negative $\forall x\in\mathbb R:$ $$x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}.$$ You can think of it as a tighter inequality than the useful $x\ge \ln(1+x)$ or $e^x\ge ...
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Laurent Series $f(z)=\frac{1}{(z+1)(z-2)}$ with conditions

I want to find the laurent series of the function $$f(z)=\frac{1}{(z+1)(z-2)}$$ for the following conditions: $|z|<1$ $2<|z|<\infty$ I have found the series for first condition, ...
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The remainder term of a maclaurin series

Suppose $f$ has a local minimum at $0$, i.e. $f'(0) = 0, f''(0)>0$, and suppose $f''$ is continuous in some interval $[0,T]$ ,$T>0$, with $f'' > 0 $ in this interval. Then we must have that $...
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Lagrange remainder - is t contained within an open or closed interval?

Above is the taylor series centred at a including the lagrange remainder. Can t be equal to a or b?
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Asymptotic expansion of integral - issue with step in proof

I have been trying to read Asymptotic Analysis by J.D Murray. There is a page in the book, rather, a certain line in this page, that has confused me thoroughly. What was said on the page before, is ...
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remainder term for a Taylor polynomial

The question asks: Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be defined by $f(x)=e^x$ (a) Find the $n$th order Taylor polynomial about x = 0, and the corresponding remainder term. I can quite ...
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Asymptotic expansion of incomplete beta function

I would like to write down an asymptotic expansion in the $N\to\infty$ limit of the following incomplete beta function $$B\left(\frac{N}{N+1};N,p+1\right)=\int_0^{\frac{N}{N+1}}x^{N-1}(1-x)^p\,\text{...
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Strange question about Taylor series

I try to get a series expansion (not necessarily Taylor) or even an integral or any nice formula, for a special continuous function satisfying a peculiar, simple functional equation: $f(x^a) = f(1+x)...
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Moment Generating Functions Taylor series

So I'm revising moment generating functions and I'm stuck on a part of a question I'm looking at. So I am asked to find the moment generating function of a random variable X whose distribution is ...
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Find the radius of the largest circle with centre at the origin inside which the Taylor series of $f$ defines an analytic function.

Let $f$ be a function that is known to be analytic in a neighbourhood of the origin in the complex plane. Furthermore, it is known that for $n ∈ \mathbb N$, $$f^{\left(n\right)}\left(0\right)=\left(n-...