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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Landau identification

Calculate taylor series for $x\rightarrow +\infty$ at the higher order allowed by the approximation present in it. $$ \sqrt{x^6+x^5-2x^3+O\left(x^2\right)} $$ I made this: $$ ...
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2answers
32 views

Tayors series exansion of $(1 - x )^n$ where $0<x<1$ and $n \ge 0$

I want to find Taylor's series or Maclaurin's series expansion of the following. $$(1 - x)^n \ \text{ where }\ \ 0 < x < 1 \text{ and }\ n \ge 0$$ will it be same as that for $$(1 + x)^n ...
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1answer
26 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
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1answer
42 views

taylor series for cosx around 0

Hey, I have the following limit, and I would like to know if it's possible to use the maclaurin series for cos(x) around 0. Is it okey to do the step I have done in the picture bellow? and let's say ...
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1answer
46 views

Taylor Theorem conceptual question

Need to solve following-- Let $$ f(x)=\begin{cases} 0& -1\le x\le0\\ x^4& 0\lt x\le 1\end{cases} $$ IF$$ f(x)=\sum_0^n\frac{f^{(n)}(0)}{n!}(x)^n + \frac{f^{(n+1)}(c)}{n+1!}(x)^{(n+1)}$$ is ...
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1answer
51 views

Evaluate the series

Let $f:(-1,1)\to \mathbb{R}$ defined as $$f(x)=\frac{x^2}{1-\cos x}$$ for $x\neq 0$ and $f(0)=2$. If $f(x)=\sum_{n=0}^{\infty}a_nx^n$ is the Taylor expansion of $f$ for all $x\in(-1,1)$, then ...
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1answer
49 views

The 7-th derivative of $ x^3 \cdot\tan(2x) $ is this right

I have to find $y^{(7)}\left(0\right)$ of $y(x)=x^3\cdot\tan{(2x)}$ So my idea was to use Taylor expansion for $\tan(2x)$ to the $7$-th element and then multiply the hole thing by $x^3 $ and then ...
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2answers
66 views

Coincidence of $x-\frac{x^3}{6}$ and $\sin x$ in an interval

Plotting $f(x)=x-\frac{x^3}{6}$ and $g(x)=\sin x$, one can see that these two function are coincide in an interval $I\subset(-\frac{\pi}{2},\frac{\pi}{2})$. On the other hand, Taylor series for $\sin ...
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1answer
50 views

Taylor series- help! [closed]

$f(x)$ is twice differentiable function in $[0,1]$. we know that: $f(0)=0$, $f(1)=1$, $f'(0)=f'(1)=0$. show that there exists a point $c$ such that $\left|f''(c)\right|\ge 4$ Thanks in advance
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255 views

Taylor series for $\cot x$

Hi guys could you show me how to do the expansion of the Taylor series of $\cot x $ at the point $x=0$. My idea was to use $\dfrac{\cos x}{\sin x} $ and I want to expand it to the second term because ...
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2answers
137 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
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2answers
43 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
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1answer
208 views

Proof of the correctness of Taylor series

I am looking at the proof provided on the wiki page for taylor series http://en.wikipedia.org/wiki/Taylor%27s_theorem#Proof_for_Taylor.27s_theorem_in_one_real_variable One of the proof provided is ...
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1answer
62 views

Mistake in Taylor expansion?

Given: The first derivative of $\tan x$ is $1/\cos^2 x$ So the derivative of $\tan x$ when $x=0$ should be $1$. This derivative times $x$ should be a term in the Taylor expansion (the term then being ...
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1answer
72 views

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges.

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges. There is an answer here that differs from mine (they claim for $-\infty<\alpha<-2$ and ...
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2answers
107 views

How to calculate Taylor expansion of $\cos(\sin x)$

I know that Taylor expansion of $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)$ and that of $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - O(x^7)$. But how do I calculte the Taylor Expansion ...
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1answer
36 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
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1answer
45 views

Taylor series expansion - application

I am working on the following: Let $f : \mathbb C \to \mathbb C$ be analytic. Suppose for all $z \in \mathbb C$ hold $f(2z) = 4f(z)$ and $f(1) = 1$. Then $f(z) = z^2$ for all $z \in \mathbb C$. I ...
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2answers
49 views

Estimate $\int_{-1}^{0}\sin(e^x)dx$ with error less than $\frac1{5000}$.

Let $f(x)=\sin (e^x)$ then the taylor polynomial of degree 2 at $x=0$ is $P_2(x)=\sin 1+(\cos1)x+\frac12(\cos1-\sin1)x^2$. I want to estimate $\int_{-1}^{0}\sin(e^x)dx$, using $P_2(x)$, with error ...
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2answers
68 views

Error estimation for $f(x)=\sin \sqrt{x}$

Let $f(x)=\sin \sqrt{x}$, then $f'(x)=\frac1{2\sqrt{x}}\cos \sqrt{x}$ and $f''(x)=-\frac1{4x\sqrt{x}}\cos \sqrt{x}-\frac1{4x}\sin \sqrt x$. Thus the Taylor polynomial of degree 2 at $x=\frac{\pi^2}9$ ...
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1answer
53 views

Laurent series for $\frac{z}{z+1}$ when $1<|z|<\infty$

Calculate the Laurent series for $\displaystyle\frac{z}{z+1}$ when $1<|z|<\infty$. There is really no singularity here, right? Can I just use a Taylor series, or what should I do?
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1answer
93 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
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33 views

Derivative of a Taylor series

I have a question about when we compute the derivative of a series. If the original series converges inside a region $R$, must its derivative also converge on the same region $R$?
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2answers
68 views

Product of two Taylor series

I have the following product of two Taylor series: $$f(x)g(x)=\frac{1}{z-1}\frac{1}{z-2}=\sum_{n=0}^{\infty} z^n \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} z^n$$ I wanted to know 2 things: 1st. How can ...
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3answers
88 views

Taylor series for $e^z\sin(z)$

How can I write the Taylor series for $e^z\sin(z)$ at $z=0$ without making the procedure too complicated? Isn't there an easier way than to compute it's derivatives and find a pattern?
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2answers
51 views

Does this limit imply that a function is “close” to Lambert W?

Suppose I am given the following limit involving function $f(n)\geq 0$: $$\lim_{n\rightarrow\infty}\log n-f(n)-\log f(n)=c$$ where $c$ is a constant. I am wondering if that implies that $f(n)$ is ...
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5answers
228 views

Taylor expansion of $e^{\cos x}$

I have to find the 5th order Taylor expansion of $e ^{\cos x}$. I know how to do it by computing the derivatives of the function, but the 5th derivative is about a mile long, so I was wondering if ...
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3answers
57 views

I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
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3answers
70 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
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1answer
64 views

Taylor series in order to find the approximate antiderivative of a function

Somewhat inspired by this question about antiderivatives, I started to check whether or not that function had an elementary antiderivative. Then, after checking with Maxima, it struck me that, by ...
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4answers
79 views

Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.

Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$. Taylor Series $$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$ I've got my ...
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4answers
77 views

$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$.

$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$. Why is the restriction $|x|<1$ or $x=1$? I know from Wikipedia that it is because out of this restriction, the ...
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0answers
46 views

Characterization of functions with fractional expansion near zero

I would like to understand if it is possible to completely characterize real-valued functions with an expansion of this type: $f(x)=f'(0)\cdot x + o(x^{\alpha})\qquad \alpha \in (1,2)$ I am not ...
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1answer
192 views

Taylor series convergence for $e^{-1/x^2}$

Consider the Taylor series for $e^{-1/x^2}$ around $0$: $$e^{-1/x^2}=1-\dfrac{1}{x^2}+\dfrac{1}{2!x^4}-\dfrac{1}{3!x^6}+\ldots$$ For which $x$ does the series on the right converge to $e^{-1/x^2}$?
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2answers
54 views

Prove $0 \le e^{-\theta x^2} \le 1$ for $0 \le \theta \le 1$

Why is it that $$0 \le e^{-\theta x^2} \le 1$$ for $0 \le \theta \le 1$? My textbook told me this in the context of langrange remainder for taylor series, and I can't figure it out. (Also, I don't ...
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1answer
67 views

calculate interval of convergence

How do I calculate the interval of convergence of $$ \frac{1+x}{1-x} $$ I made it into a taylor series expansion using first principles and the sum is this $$\sum_{n=-\infty}^\infty \left( \{ ...
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1answer
50 views

Odd nature of sine function. (Taylor series)

Although, this might be silly question. I am just wondering what happens to the odd nature of $\sin \theta$ when I expand it about some $ \pi/4 $. There are terms with even powers appearing as well. ...
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1answer
84 views

Taylor (Maclaurin) Series remainder for ${\rm sin}\ (x)$

So I just finished doing this problem and I think the solution I got is wrong, it seems a bit too large. According to my calculations, I need 36 terms. I fear I've made a mistake and I would really ...
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4answers
60 views

Taylor series of $(1+x)\ln(1+x)$ in $x=0$

How to determine the Taylor series of $(1+x)\ln(1+x)$ in $x=0$? My idea is finding the second derivative of the expression, which is $\frac{1}{1+x}$. The Taylor series of this expression is ...
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1answer
52 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
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3answers
272 views

Evaluating $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
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3answers
157 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
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2answers
49 views

Please help me understand Rudin Theorem 5.15

I am having trouble understanding the intuition behind the last part of this theorem. I'd appreciate some help understanding the intuition behind the last equation: $f(\beta ) = P (\beta ) + ...
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2answers
41 views

Estimating error in Taylor polynomial

Consider the nth order Taylor polynomial for cos x centered at 0 dented T(n) (x,0). How larger must we take n to guarantee that the error |cos x-T(n) (x,0) |is at most 10^-3 for x in [-pi/2,pi/2)
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2answers
82 views

question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
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0answers
38 views

Inequality sine power series (induction)

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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1answer
66 views

Linearize a simple ODE

This is homework. I have $\displaystyle \qquad S\frac{dh(t)}{dt} + \frac{1}{R}\sqrt{h(t)} = q(t)$ and need to linearize it. Setting all derivatives to zero, I get the steady-staty value of $h - ...
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2answers
50 views

What is the characteristic function used for?

Im totally new to statistics , but what is the characteristic function for ? I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the ...
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1answer
50 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
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0answers
70 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...