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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0answers
361 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
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6answers
5k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
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0answers
33 views

Solution of an equation with a taylor expansion

Show that $$x=ne^{-x}$$ with $n \in \mathbb{N}$ and $n\ge1$ has one solution $x_n$ such that: $$x_n = \log(n)-\log(\log(n)) + o(\log(n))= \log(n)-\log(\log(n)) + o(1)$$ My try: Let be ...
4
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1answer
3k views

Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form ...
0
votes
1answer
23 views

Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
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2answers
30 views

Inequality doubt with taylor expansion

Can I prove that $\forall x>0$ $$e^{x/(1+x)} < 1+x$$ Showing that $e^{x/(1+x)} = 1+x-\frac{x^2}{2}+o(x^2)$ and so $-\frac{x^2}{2}+o(x^2)<0$ for all $x>0$? How i can be sure that $o(x^2)$ ...
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2answers
62 views

Taylor expansion of $\frac{1}{2-z-z^2}$

The problem is: Find the Taylor expansion of $f(z):= \dfrac{1}{2-z-z^2}$ on the disc $|z| < 1$ So far I have used partial fractions to obtain $f(z) = \dfrac{1}{3}\left(\dfrac{1}{1-z} + ...
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0answers
19 views

General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
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2answers
169 views

Find the power series for $f(x) = \frac{\cos(x^3)}{2x^2}$

I'm pretty sure if it were just $\cos(x^3)$ i could subsititue $x^3$ for $x$, everywhere in the known series, but what do I do because it's divided by $2x^2$?
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votes
2answers
466 views

Prove Euler's Formula using MacLaurin Series

How can you prove $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ (Euler's Formula) using MacLaurin Series? Thanks!
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1answer
43 views

Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
2
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2answers
116 views

Taylor Series looks like exponential

guys! I need a little help. I have this series $$ \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k $$ where $j \in \mathbb{N}$. I need to know if the limit of this function when $t$ goes to ...
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vote
1answer
53 views

MacLaurin powerseries and interval of convergence

Given the function $f(x) = 5/(6*x^2-x-1)$, (a) Expand into MacLaurin powerseries the function $f$ up to order $3$. (b) Find the interval of convergence of it. (a) I will use the type of ...
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votes
2answers
69 views

Geometric Series into Maclaurin Series

Expand 1/(1+x) into Maclaurin Series I found f(0)=1, f'(0)=1, f''(0)=2!, f'''(0)=3! and so on Therefore f^(k)(0)=k! so would the series centered at 0 be equal to x^k ? Just want to check to see if I ...
0
votes
2answers
87 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
2
votes
2answers
142 views

Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
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1answer
69 views

Find the power series for $d/dx(\arcsin x)$

How would you find the general power series for $\frac{1}{\sqrt{1-x^2}}$ , without using the general rule for arcsinx? I understand it is necessary to use binomial series, but I am having trouble ...
2
votes
3answers
143 views

Taylor Theorem inequality

Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
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0answers
73 views

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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0answers
158 views

Maclaurin series of $\sin(2\pi x)$

Find the Maclaurin Series for $$f(x) = \sin ( 2 \pi x )$$ using the definition of a Maclaurin series. If $f(x) = \sum_{n=1}^\infty c_{2n+1}x^{2n+1}$, give $c_{2n+1}$: ...
2
votes
1answer
261 views

taylor series expansion about undefined point

Question is to find coefficient of $(z-\pi)^2$ in taylor series expansion about $\pi$ of $$f(z)=\frac{\sin z}{z-\pi},z\neq\pi$$ $$=-1,z=\pi$$ now, coefficient of $(z-\pi)^2$ should be ...
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1answer
99 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
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2answers
170 views

Solving a limit using MacLaurin series

I want to find $$ \lim_{x\to0} \frac{(e^{-x^2}-1)\sin x }{x\ln(1+x^2)}$$ using a Maclaurin series and not using the l'Hôpital's rule. However I can't seem to get it right. Thanks for any possible ...
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0answers
38 views

Taylor Series on two variables simultaneously

I need to find Taylor series of u($x_i+h,t_n+k)$ where taylor series of $u(x_i,t_n+k)$ and $u(x_i+h,t_n)$ are given as such. Is it basically the summation of both the terms except $u(x_i,t_n)$? ...
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votes
2answers
54 views

Application of Taylor's Formula

If we are given that $f''(x) = f(x)$, how do we show that there exist constants $a$ and $b$ such that $f(x) = ae^x + be^{-x}$ for all $x$? A hint is given: We can define another function $g$ by $g(x) ...
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2answers
66 views

Power series of a function about a non zero point

No clue how to ask questions here so here goes nothing! How do I work towards finding the power series of a function centered about a point a not equal to $0$? The specific question I was asked is to ...
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1answer
556 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
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0answers
90 views

Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
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2answers
58 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
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vote
1answer
91 views

differentiate arctan (maclaurin?)

I have this assignment: Differentiate this expression: $$ f(x) =\arctan \frac{x-1}{x+1} $$ There is also known that $-1 < x$ (Why is that important?). I do not know how to solve this problem... ...
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2answers
63 views

Solve this limit (Maclaurin or differentiate?)

I have this assignment where I should calculate the limit below: $$ \lim_{x\to0}\frac{\sin 2x}{x\cos x} $$ I can use l'Hospitals rule (because it is a "zero divided by zero"-case) and therefore ...
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1answer
44 views

How Hessian feature detector works?

I know about Harris corner detector, and I understand the basic idea of its second moment matrix, $$M = \left[ \begin{array}{cc} I_x^2 & I_xI_y \\ I_xI_y & I_y^2 \end{array} \right]$$, edges ...
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1answer
355 views

Base 2 logarithm with Taylor expansion

I'm trying to implement the natural logarithm in C, and our task is to make it really efficient. So what we are doing is, that we use the first 8 members of the series. This works fine, but the ...
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2answers
44 views

Doubt about Taylor's polynomial to approximate f(x)

Use the degree 2 Taylor polynomial of $f(x) =$ $\sqrt[3]{1728 + x}$ to approximate $\sqrt[3]{1731}$ and give a bound for the error. To obtain the degree 2 Taylor's polynomial, I computed the second ...
0
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1answer
91 views

Taylor Approximation

I have got this question Suppose that $f$ is twice differentiable at every $x\in\mathbb{R}$ and that for every $x\in\mathbb{R}$ $$f''(x) + f(x) = 0.$$ Show that if $f(0) = 0, f'(0) = 0, $ and ...
3
votes
2answers
73 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
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0answers
47 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
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votes
2answers
153 views

Maclaurin polynomial for $\arcsin(x)$

How would I find the 3rd-order Maclaurin polynomial for $f(x) = \arcsin(x)$; with the interval $(0,\frac 3 4)$ to show it in terms of $x$? Would you have to somehow manipulate it to $\dfrac{1}{1+x}$ ...
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vote
1answer
43 views

Is this series G(1/n) convergent or divergent given G(x)?

Suppose $G(x)=\int_0^x\sin{\left(e^s-1\right)}ds$ Does the series $\sum_{n=1}^{\infty}G(\frac{1}{n})$ converge or diverge? I'm not sure how to go about solving this; however in our notes it says ...
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3answers
37 views

Question on a Taylor Polynomial

We are asked to generate the taylor polynomial $P(x)$ for $$ f(x) = \frac{e^{{(x-1)}^2}-1}{(x-1)^{2}} $$ about $x=1$ Using substitution into the known taylor polynomial of $e^{x}$ and further ...
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votes
2answers
48 views

Taylor polynom, residual for symmetric values

When creating the taylor polynom for a $C^3$-function around a certain point i get the formula $f(z+h)=f(h)+hf'(z)+\frac{h^2}{2}f''(z) + \frac{h^3}{6}f'''(z) + R$ Now lets say I create the polynom ...
3
votes
1answer
96 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
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3answers
281 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
2
votes
6answers
315 views

Definition of matrix exponential

Is there an alternative definition of a matrix exponential so I can use it to prove that $$e^{A}=\sum_{m=0}^{\infty} \frac{1}{m!}(A)^m \;?$$ Thanks a lot in advance!
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votes
1answer
201 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
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2answers
102 views

Power Series Proof

I understand that given that expanding a function onto polynomials is a valid thing, the equation for Taylor series follows, but why is expanding a function onto polynomials a valid thing to do? Why ...
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5answers
237 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
0
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1answer
42 views

Taylor Expansion and Log transformation (Time Series)

From: Time Series Analysis with Applications in R by Jonathan D. Cryer and Kung-Sik Chan. Here is the Taylor expansion: $\log Y_t = \sum_{n = 1}^{\infty} (-1)^{n+1} \frac{(Y_t - 1)^n}{n} $. How ...
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2answers
47 views

what is the Maclaurin Series of this function?

Can anyone explain to me how to find the Maclaurin series of: $$f(x)=(x^2+1)e^{\frac{-x^2}{4}}$$ and why does it converge for every x? thanks,