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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3
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1answer
65 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 ...
0
votes
3answers
425 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
3
votes
2answers
531 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 ...
3
votes
1answer
80 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 ...
0
votes
1answer
53 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 ...
1
vote
1answer
66 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 ...
2
votes
1answer
69 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?
1
vote
2answers
62 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 ...
1
vote
2answers
64 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 ...
3
votes
1answer
197 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 ...
0
votes
2answers
87 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 $x=\frac{\pi^2}9$ ...
2
votes
2answers
165 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 ...
1
vote
2answers
37 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 $R$?
1
vote
3answers
167 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?
1
vote
0answers
116 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
2
votes
1answer
100 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 ...
0
votes
3answers
58 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. ...
1
vote
3answers
77 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 ...
2
votes
4answers
110 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 ...
2
votes
4answers
82 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 ...
3
votes
0answers
54 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 ...
3
votes
1answer
851 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 $e^{-1/x^2}$?
8
votes
3answers
428 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 ...
2
votes
1answer
69 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 - ...
1
vote
2answers
55 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 ...
0
votes
1answer
381 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( \{ ...
4
votes
3answers
130 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
4
votes
1answer
57 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. ...
0
votes
1answer
84 views

Second partial derivation of vector function and taylor series

I have vector function: $$ f(x,y) = \begin{pmatrix} (R+r\cos(y))\cos(x) \\ (R+r\cos(y))\sin(x) \\ r\sin(y) \end{pmatrix} $$ I have done Jacobian of that function: $$ f'(x,y) = \begin{pmatrix} ...
2
votes
1answer
117 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 ...
3
votes
4answers
71 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 ...
1
vote
1answer
73 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 ...
1
vote
3answers
405 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 $0$?
3
votes
2answers
149 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 ) + ...
1
vote
2answers
53 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)
0
votes
1answer
30 views

Maclaurin series and expressing as a ln(argument)

Found this question in my old homework notes that I did not do at the time! I always wondered how I do this... The first part is a explanation. It is kind of long. Sorry! Here is the actual ...
1
vote
2answers
91 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?
0
votes
2answers
55 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 ...
0
votes
4answers
153 views

Infinite series expansion of $e^{-x}\cos(x)$

Establish an infinite series expansion for the function $y=e^{-x}\cos(x)$ from just the known series expansions of $e^x$ and $\cos(x)$. Include terms up to the sixth power. I know that the ...
2
votes
0answers
82 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 ...
2
votes
1answer
131 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 ...
3
votes
1answer
57 views

Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
1
vote
1answer
45 views

f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor ...
1
vote
0answers
30 views

Find the order of the following expression as x->0

Could someone help me find the order of the following expression without using the quotient rule? $\frac{1-\cos(x)}{1+\cos(x)}$ I expanded the denominator and the numerator but not sure how I get to ...
2
votes
2answers
1k views

Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
0
votes
4answers
128 views

Find $a$ such that $\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3$.

Find $a$ such that $$\lim_{x\to 0} \frac{1-\cos(\sqrt{ax})}{x^2}=3.$$ Can we solve it with l'Hospital's Rule or do we need to use Taylor series? I have tried using L'Hospital's Rule and i keep ...
2
votes
1answer
38 views

Proving that 2 functions are equal/not equal

Prove the equality of $f_1$ and $f_2$ given the following conditions: Problem 1 $f_1(x)$ and $f_2(x)$ are functions of finitely summed sine and cosine functions (e.g. $3\cos2x+\sin5x$), any ...
1
vote
1answer
45 views

Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
8
votes
1answer
132 views

How can I compute this limit? [duplicate]

I have to compute $$ \lim_{n\to\infty} \exp(-n)\left(1+n+\frac{n^2}{2}+\ldots+\frac{n^n}{n!} \right)$$ I think the value is 1, but i don't know how to proof this. Do I have to estimate the remainder ...
2
votes
0answers
81 views

Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ ...