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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1answer
59 views

Laurent-series expansion of $\frac{1}{(e^z-1)^2}$ about $z=0$

I am studying for exams in complex analysis and taking a look at past papers. This comes up often or an integral of the given function along a certain curve, which is actually the same thing since the ...
3
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4answers
56 views

Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . [closed]

Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . Express your answer in sigma notation, simplified as much as possible. This is a practice question that I am having trouble ...
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1answer
18 views

sin(x+y^2) taylor expansion little oh error term degree >3

I am trying to understand example 3.4.5 in John and Barbara Hubbard's second edition of Vector Calculus, Linear Algebra, and Differential Forms. It provides the taylor expansion of $sin(x+y^2)$ by ...
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1answer
74 views

Taylor series expansion?

How to find the Taylor series expansion of $$(1+x)^{1/x}$$ I tried with the Taylor series but unable to solve it. Help me out. Hints or anything that sort will be helpful.
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2answers
61 views

Taylor expansion of logarithm function.

Expand $f(x) = \log(1 + x)$ around $x = 0$ to all orders. More precisely, find $a_n$ such that for any positive integer $N$, we have$$f(x) = \left(\sum_{n=0}^{N-1} a_nx^n\right) + E_N(x) \text{ for ...
4
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1answer
70 views

How to compute $\lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a function such that : $f(x)=x-x^3+o(x^3).$ Compute $$ \lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$$ My thoughts: ...
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1answer
84 views

Behaviour of $f(x)=e-\left(1+\frac{1}{x} \right)^{x}$ when $x\to+\infty$

This is from an MCQ contest. For all $x\geq 1$ let $f(x)=e-\left(1+\dfrac{1}{x} \right)^{x}$ then we have : $f(x)\mathrel{\underset{_+\infty}{\sim}}\dfrac{e}{x}$ and $f$ is integrable on ...
3
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0answers
37 views

Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
3
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1answer
38 views

Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
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1answer
118 views

Taylor expansion for $\arcsin^2{x}$

I stumbled upon this particular expansion that was included in this post. $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$ This caught ...
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0answers
123 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry ...
2
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1answer
38 views

When is $a(z) = b(c(z)) $?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
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0answers
26 views

Central Difference taylor approximation

We are asked to show that We have so far managed to show the first two equalities using finite difference approximations but the last one still eludes us. Any hints?
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0answers
25 views

Generalised Taylor series to fractional order derivatives and special functions

A year ago or so I read this papar which was wonderfully illuminating link. For example the author seduces the reader with wonderfully compact representations like that of the bessel $J_v(z)$ function ...
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1answer
57 views

Looking for Taylor series expansion of $\ln(x)$

We know that the expansion of $$\sin(x) $$ is $$x/1!-x^3/3!\cdots$$ Without using Wolfram alpha, please help me find the expansion of $\ln(x)$. I have my way of doing it, but am checking myself with ...
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2answers
73 views

Coefficients of series

Suppose that i have a function $f(x)=\sum_{i=0}^{\infty}a_ix^i$ with radius of convergence $r_f>0$ and that i want to write $f$ in a form $f(x)={e^{g(x)}}$ where $e$ is natural logarithm base and ...
1
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1answer
28 views

Taylor expansion $\frac{y-(y+1) \log (y+1)}{y^2 (y+1)}$

I would like to use Taylor expansion around $y=0$ for that expression but i got stuck $$\frac{y-(y+1) \log (y+1)}{y^2 (y+1)}$$ Taylor expansion around $y=0$ ...
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5answers
64 views

Maclaurin expansion of $y=\frac{1+x+x^2}{1-x+x^2}$ to $x^4$

Maclaurin expansion of $$\displaystyle y=\frac{1+x+x^2}{1-x+x^2}\,\,\text{to } x^4$$ I have tried by using Maclaurin expansion of $\frac1{1-x}=1+x+x^2+\cdots +x^n+o(x^n)$, but it seems not lead ...
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0answers
47 views

Maximize a function relative to a constraint. The question is related to the Taylor series for the cosine function

Let $f: \mathbb{N}_0 \to \mathbb{R}^+$ be given. Assuming there exists a function $g : \mathbb{N}_0 \to \mathbb{R}^+$ such that $g(n) \geq f(n)$ for all $n$ and $$ D(f, g) = \sum_{\substack{n=0\\ n ...
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3answers
63 views

An infinite sum in the product of sines

This is an undergrad or lower level question I need help with. Evaluate $$\quad \sum_{n=1}^{\infty} \sin{\left(\frac{a}{3^n}\right)}\sin{\left(\frac{2a}{3^n}\right)}$$ where a is just some real ...
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1answer
32 views

Is this Taylor series correct taken correctly? Confused reasoning

I have $dx/dy=-ay, x(0)=1$ initial value problem. Then $x(y)=\frac{x(0)}{0!}y^0+ \frac{x'(0)}{1!}y+\frac{x''(0)}{2!}y^2=1+(-a)y+a^2y^2...$
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1answer
98 views

Intuition behind power series

I keep seeing power series throughout mathematics disguised in all different shapes, yet I can't seem to put my finger on what is really fundamentally being expressed here. Some examples: Arabic ...
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2answers
25 views

Find Maclaurin expansion of $y=2^x$ to $x^4$

Find Maclaurin expansion of $$y=2^x\text{ to } x^4$$ This is my try. We have $\displaystyle 2^x=e^{x\ln 2} =\left[1+\frac{x^2}2+\frac{x^3}6+\frac{x^4}{24}+o(x^4)\right]^{\ln 2}$ with $o(x^4)$ is ...
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0answers
21 views

Derivative of a definite improper integral

-The derivative with respect to beta, for the following definite integral is required. g = $\int_\beta^{\sqrt(\beta^2 +1}$ $erfc(\gamma z)/\sqrt(z^2 - \beta^2)$dz -I am using the leibniz formula ...
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1answer
111 views

Solving initial value problem with Taylor Series expansion $dx/dt=x^2, x(0)=1$

I have series of homework questions that call for using Taylor series to solve initial value problems so can someone solve this example and explain what they are doing? I also want to know what ...
2
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1answer
2k views

What is the nth derivative of $\dfrac{1}{\sqrt{1 + x^2}}$

I'm trying to find a general formula for the $n$th derivative of $$\dfrac{1}{\sqrt{1 + x^2}}$$ I got up to, \begin{eqnarray*} g^{(0)}(x) &=& g(x) \\ g^{(1)}(x) &=& \dfrac{1}{(1 + ...
2
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1answer
74 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ . [duplicate]

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
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2answers
85 views

Lagrange Remainder for an general binomial series

We started this somewhat in class, but I wanted some more explanation in a general scenario. Given a binomial series $$(1+x)^{a} \approx 1+\alpha x + \frac{\alpha(\alpha -1)}{2!}x^{2} + \frac{\alpha ...
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0answers
64 views

Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set ...
2
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0answers
29 views

expansion of Bessel function $J_1$

this is a general question. Is there a general way to expand the Bessel Function $J_1(z)$ when $z\in \mathbb{C}$ and when z is large? Or in other words, what is the asymptotic expansion of $J_1(z)$? ...
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0answers
27 views

If $y'(x) =y(x)-x +1$ and $y(0)=1$ then $y^{(n)}(0) = 1$ for every $n\ge2$

Start of the taylor series: f(x) = f(0) Which from the question is 1. f(x) = 1 f'(0) = y - 0 + 1 The current y value ...
6
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2answers
15k views

What's the Maclaurin series for $\arcsin(x)$?

I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake? $$ ...
0
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1answer
25 views

Taylor series expansion and radius of convergence

The problem is: Expand the given function using Taylor's expansion around $a=1$. $f(x)=(5x-4)^{-7/3}$, and then find the radius of convergence of the obtained series. Hint : Write the nth derivative ...
0
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1answer
183 views

Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of ...
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1answer
21 views

Limit Question involving logarithmic taylor expansion

I need to evaluate the limit for part of my proof: $$ \lim_{n \to \infty}\left(1-\dfrac{1}{\eta^{x}} \right)^n $$ My attempt: \begin{align*} \lim_{n \to \infty} F_{\eta_n}(x) &= \lim_{n \to ...
3
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3answers
89 views

How to show $\sum_0^\infty \frac{x\lambda^x} {x!} = \lambda e^\lambda$?

I know that $\sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda$, but I'm having a really difficult time dealing with the extra $x$.
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2answers
154 views

Taylor Series Expansion/Small Angle Approximation

I'm working on my first physics problem set (mainly math review) and I'm having a really hard time with one of the questions: Trigonometry: Start with the general expression: $f(x)=a+bx+cx^2+ ...
0
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1answer
74 views

Expression for the gradient using Taylor's Theorem

I've just started reading Nocedal and Wright's book on Numerical Optimization. On page 14 there is a formula for the value of the gradient in some point (equation 2.5) that I cannot derive myself. ...
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2answers
44 views

Taylor expansion of a random variable

I'm struggling a little with this expansion: Where $E$ is the expectation operator, $U$ is a function of $Y$ and $Z^~$ is a random variable. In the second passage why the expansion looks like ...
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2answers
91 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
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0answers
30 views

Deriving Taylor theorem expression

In one book, I've got a following written: Substituting for $f′(x)$ in (4.15), we obtain the second approximation: $$f(a +h) \approx f(a) + \int_a^{a+h}[f'(a) + (x-a)f''(a)]dx$$ $$f(a ...
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0answers
24 views

Big Oh symbol in Taylor expansions

Consider a remainder of some Taylor series: \begin{align}\frac{Mx^6}{C} + \frac{M'x^8}{C'} + \frac{M'' x^{10}}{C''} + ...\end{align} I want to replace this with $\mathcal{O}(x^\alpha)$ for the best ...
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0answers
17 views

Taylor series finding approximation within a inteval

Let $f(x)=(1-x)^{-1}$ and $x_0=0$. Find the nth taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$ Find a value of n needed for $P_n$ to to approximate $f(x)$ to within $10^{-6}$ on the interval ...
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0answers
54 views

Relationship between Laplacian and Taylor expansion for 2nd derivative

I am working on converting a solution to a certain PDE from working on a regular 2D grid to work on a 3D triangular mesh. In the 2D scenario the 1st and 2nd derivatives are, of course, approximated ...
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3answers
118 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I don't understand it and I give up the math. 10 year ago, I needed to work with calculus during work and this question came to ...
3
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1answer
39 views

Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form: $$\sum_{n=1}^k ...
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1answer
73 views

Polynomial approximation for $f$ induces an approximation to $\sqrt f$?

Assume $f:[0,1] \rightarrow \mathbb{R}$ satisfies $f(t)\geq 0, f(0)=0$ I am looking for a machinery, which given a polynomial approximation of $f$ of a certain degree, determines the highest order ...
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2answers
78 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
0
votes
1answer
16 views

Taylor series expansion for $g(t+k,u(t+k))$

I am working on predictor corrector schemes for parabolic PDEs and in my derivations I had to find the Taylor series expansion for $g(t+k,u(t+k))$ where $g$ is a function of $t$ and $u$, $u$ is a ...
0
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1answer
284 views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem. For what range of values of $x$ will this ...