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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3answers
51 views

How do you write the taylor series for (arctan(x))^2

I know that the taylor series of $\arctan{(x)}$ is $x - \frac{x^3}3 + \frac{x^5}5 + \dotsc$ In order to square it I would have to multiply it by itself. $(x - \frac{x^3}3 + \frac{x^5}5 + \dotsc) \...
-1
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1answer
27 views

Taylor expansion with geometric series [closed]

Expand $1/x $ a taylor series around $x=1$. Use $\frac {1}{1-(1-x)}$ and used the geometric series. I know $\sum (1-x)^n = \frac {1}{1-(1-x)}$, $-1 <x -1<1$
2
votes
1answer
113 views

Finding an upper bound on the Taylor remainder

Any help with the proof I have posted for the following question is greatly appreciated; I have listed my particular issue at the end. Thank you! Let $p \geq 0$. For $n=1,2, \dots,$ find $P_n(x)$ ...
1
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0answers
57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
0
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1answer
43 views

Why can't I get good approximation when choosing values away from point of expansion? (Taylor series)

I was in the middle of doing a computing project assigned to me when I came across the question. $\operatorname{P}_N(x)$ is the taylor polynomial for $f(x)=\ln(x)$ expanded around pouint $x_0=1$ $$\...
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0answers
39 views

Find the cubic Maclaurin polynomial for $\sinh x$ and estimate the remainder on the interval $|x| \le 1 $

I'm working through the above problem, and am having trouble deciphering what it means by "estimate the remainder on the interval $|x| \le 1$". I found the Maclaurin polynomial to be $$P_3(x) = x + \...
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1answer
32 views

Taylor's Theorem for 2D function

I am doing a numerical analysis course and we are looking at numerical methods for solving initial value problems. For example: methods such as $y_{n+1} = y_n+\frac{h}{2}(f(t_n,y_n) + f(t_{n+1}, y_{n+...
1
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1answer
39 views

Proofs for Taylors theorem and other forms

Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$, $$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$ and after this use this ...
4
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0answers
178 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
1
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1answer
36 views

how to find taylor serie for 1/z with |z| > 0?

I have the following and I need to give the Laurent development for |z| > 0. The Laurent development in this form : and to give few a(n) coefficients How can it be done? normally we use the ...
0
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2answers
30 views

Taylor series, identify radius of convergence

I have the following function : I need to find it's radius of convergence with z0 = 0. The function is analytic everywhere except where 1 + sin(iz) = 0 (to my ...
0
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1answer
68 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 ...
0
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1answer
20 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 ...
1
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1answer
76 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.
2
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2answers
63 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 $$f(x)\mathrel{\underset{_+\infty}{=}}(1-a)+\dfrac{1-b}{x}+...
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 ...
8
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1answer
132 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 ...
5
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0answers
125 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 $x,y&...
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 ...
0
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0answers
29 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?
0
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0answers
26 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
59 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 $y=0$ $$\log(1+y)=y-\frac{y^2}{2}+\frac{y^3}{3}+...
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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 me ...
0
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0answers
50 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
64 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 ...
1
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1answer
33 views

Is this Taylor series correct taken correctly? Confused reasoning

I have $dx/dy=-ay, x(0)=1$ initial value problem. Then $x(y)=\frac{x(0)}{0!}y^0+ \frac{x'(0)}{1!}y+\frac{x''(0)}{2!}y^2=1+(-a)y+a^2y^2...$
2
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1answer
108 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 ...
0
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0answers
22 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
123 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
76 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 ...
0
votes
2answers
90 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 (\...
2
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0answers
70 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 ...
7
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2answers
16k 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
27 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
202 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 $t$....
0
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1answer
22 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
votes
3answers
102 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 $x$.
1
vote
2answers
160 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
votes
1answer
77 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. ...
1
vote
2answers
45 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 what'...