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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Taylor expansion: $\lim\limits_{x\to 0}{\frac{\sqrt{x}\sin{\sqrt{x}}+\log(1-x)}{x-\tan{x}}}$

Could someone tell me if my proceeding is correct? $$\lim_{x\to 0}{\dfrac{\sqrt{x}\sin{\sqrt{x}}+\log(1-x)}{x-\tan{x}}} =$$ $$= \lim_{x\to ...
4
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2answers
448 views

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor's Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How ...
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1answer
73 views

Calculating Taylor polynomials for a function in $\mathbb{R}^3$.

Hello how does one apply the Taylor polynomials in a function of three variables? If we consider the function $f: \mathbb{R}^3$ to $\mathbb{R}$ with $(x_1, x_2, x_3)$ mapped to $\sin(x^2_1) + ...
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2answers
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Asymptotic Expansion

I was trying to solve and ODE and while doing some asymptotics, I bumped into something like this $\left(1+\frac{\gamma}{z_{0}}+\epsilon \frac{z_{1}}{z_{0}}\right)^{-2}$ where $\gamma$ $\,$ is of ...
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127 views

A recursively defined sequence

My question: How does one determine the dependence upon $n$ of the sequence $(c_n)$ uniquely (well-)defined by $$ c_0 = 1~~, $$ $$ c_1 = \frac{1}{2}~~, $$ $$ \sum_{k=0}^m(m-k+1)c_kc_{m-k+1} = 0 ...
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4answers
233 views

Why do the endpoints of the Maclaurin series for arcsin converge?

The series $$\sum_{n=0}^\infty {{-\frac {1} 2} \choose n} \frac{(-1)^n}{2n+1}$$ is an endpoint for the Maclaurin series for arcsin(x). (The other endpoint is just the negative of this one.) I played ...
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1answer
96 views

$1 - (-k \lambda) + (k\lambda)^2 /2 = (k\lambda +O(k\lambda^2))$?

I'm trying to follow some notes my supervisor has written and I've got the first three terms of a Taylor series $$1 - (-k \lambda) + (-k\lambda)^2 /2$$ becomes $$k\lambda +O(k\lambda^2)$$ Is this ...
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3answers
164 views

Proof an inequality

I'm trying to prove that $$ \frac{3-2\sqrt{1-15 m^2}}{1+12 m^2}\geq 1+3 m^2$$ I have obtained in a CAS software the Taylor expansion in $m=0$ One posibility to prove the inequality is showing ...
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1answer
328 views

Using binomial theorem find general formula for the coefficients

Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion: $$ ...
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2answers
483 views

Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?

When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way? Also, is there ...
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0answers
295 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
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1answer
32 views

How large need $n$ be taken to ensure that $T_n(x)$ gives a value of $\ln(1.3)$ which has an error of less than $0.0002$?

$$ f(x) = \ln(1+x)$$ The previous part of this question required me to write down the remainder term for the taylor polynomial of order n. My remainder term worked out to be: $$R_n(x) = (-1)^n ...
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1answer
207 views

What is the Taylor series for $g(x) =\frac{ \sinh(-x^{1/2})}{(-x^{1/2})}$, for $x < 0$?

What is the Taylor series for $$g(x) = \frac{\sinh((-x)^{1/2})}{(-x)^{1/2}}$$, for $x < 0$? Using the standard Taylor Series: $$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}$$ ...
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2answers
101 views

Taylor's formula

Taylor's Formula Write taylor's formula for $F(x,y)= \sin(x)\sin(y)$ using $a=0$, $b=0$, and $n=2$. $$\sin(h)\sin(k)=hk−\frac 16h(h^2+3k^2)\cos\theta h\sin\theta k−\frac 16 k(3h^2+k^2)\sin\theta ...
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3answers
344 views

Remembering Taylor series

Could anyone suggest a good way of memorizing Taylor series for common functions? I have tried to remember them but never seem to be able to commit them to permanent memory.
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5answers
185 views

Can every polynomial be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 +\ldots + a_n (x-x_0)^n$

Can every polynomial of degree $n$ be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \ldots + a_n (x-x_0)^n$ while $x_0$ is an arbitrary but given real number and all $a_k$ can be freely chosen? Is ...
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1answer
161 views

Asymptotic of Taylor series

Let $f(x)$ and $g(x)$ be two Taylor series such that: $$ f(x)= \sum_{n=0}^{\infty}(-1)^{n} a(n) x^{n} $$ and $$ g(x)= \sum_{n=0}^{\infty} b(n) x^{n} $$, for $ a(n) >0 $ and $b(n) > 0 $. My ...
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2answers
121 views

What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$?

I had this question on a quiz and I answered 35!x^5/4! What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$? Use a suitable Taylor Polynomial for $e^x$ at $x = 0$. Express the answer in ...
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1answer
1k views

Taylor expansion around infinity of a fraction

I want to Taylor expand the following function: $f(x)=\frac{1}{a-x^2}$ when $x \rightarrow \infty$. I know the result (from Wolfram Alpha) to be $-\frac{1}{x^2} - \frac{a}{x^4} + O(\frac{1}{x^6})$ ...
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1answer
86 views

Equivalent of $ \int_1^x e^{-\sqrt{\ln(t)}} \mathrm dt $ when $x \rightarrow \infty$

How can I prove that: $$ \int_1^x e^{-\sqrt{\ln(t)}} \mathrm dt \sim_{x \rightarrow \infty} xe^{-\sqrt{\ln(x)}}$$ without using l'Hôpital's rule ? Integration by parts: $$ \int_1^x ...
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1answer
800 views

Difference in limits because of greatest-integer function

A Problem : \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} results in the solution : 1 But the same function enclosed in a greatest integer function results in a 0 ...
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1answer
270 views

a multivariate quadratic function

Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where $$f_1(x,y)= x^2+3xy$$ $$f_2(x,y)= 2xy+y^2$$ (here f is column vector, x, y are variables) Assume that each $f_i$ is a ...
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2answers
122 views

Need a hint with this question

I'm looking over one of my past papers and I'm having some trouble with the following question. By considering the series expansion of: $\ln(1-z)$, where $z=\frac{e^{i\theta}}{2}$, show that ...
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3answers
533 views

Prove that the Taylor series converges to $\ln(1+x)$.

Prove the following statement. For $0 \leq x \leq 1$, the Taylor Series, $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ converges to $\ln(1+x)$ Any help will be greatly appreciated! ...
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1answer
954 views

Taylor expansion for matrices

Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
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1answer
92 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
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111 views

Problem regarding infinite sum of remainders.

Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly): Find ...
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3answers
348 views

Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
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1answer
245 views

Power Series Definition

What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
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3answers
712 views

Showing that the Taylor series of $f(x)=\frac{1}{\sqrt{1+x}}$ converges to $f$ on (I think) the interval (-1,1].

Let $f(x)=\frac{1}{(1+x)^{1/2}}$. Supposed to find the Taylor series $Tf$ of $f$ around $0$, and show that it converges to $f$ on $[-1,1)$, (although I suspect there's a misprint in the book, and that ...
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2answers
795 views

Taylor Series of Products

Here's a taylor series problem I've been working on. I'll list a few steps to the problem and tell you guys where I'm getting stuck. Thanks in advance for the help. So my questions builds off the ...
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2answers
434 views

Multiplication of Taylor series - expanding $2x\sin(x)$

I'm working on a problem for university Calculus 2. We're talking about Taylor series right now and I need to approximate an integral using one of a function that I think it should be easy to produce ...
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1answer
2k views

Taylor Expansion of Error Function

I am working on a question that involves finding the Taylor expansion of the error function. The question is stated as follows The error function is defined by $\mathrm{erf}(x):=\frac ...
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1answer
376 views

Bounding Error term in Taylor Expansion of $\sqrt{1+x}$

I am attempting to justify the expansion $$ \sqrt{1+x}= 1 + \frac{x}{2} + \sum_{n=2}^{\infty}{(-1)^n \frac{1}{2n}\frac{(1-\frac{1}{2}) \cdots ((n-1)-\frac{1}{2})}{(n-1)!}x^n} $$ for $-1<x\leq 1$ ...
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1answer
261 views

Convergence radius of the solution of an ODE.

Sorry for the vague title... Here's the problem: Consider the ODE $$y'' + \frac{1}{\sqrt{t}}y=0.$$ Given a solution $y$ such that $y(t_0)=0$ for a fixed $t_0>0$ and $y'(t_0)\neq 0.$ What can i say ...
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2answers
2k views

Quadratic approximation of a function

Here en.wikipedia.org/wiki/Taylor_theorem i have found that linear approximation of f at the point a is $$P_1(x)=f(a)+f'(a)(x-a) $$ For the quadratic approximation the quadratic polynomial is ...
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1answer
101 views

Limit of a sum for which the upper limit is also in the argument of the sum - Taylor series of $e^x$

A book I'm reading claims that $\frac{1}{2}(k-1)!\sum \limits_{j=0}^{k-3} \frac{k^j}{j!} \sim (\pi / 8)^{1/2}k^{k-\frac{1}{2}}$ as $k \to \infty$. I can get most of the expression to work out nicely ...
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4answers
199 views

Finding a three-term asymptotic expansion of the inverse function of $ f(x)=x^3+x$

I would like to find a three-term asymptotic expansion of $g$ the reciprocal function of: $$ f(x)=x^3+x$$ We have: $$ f(g(x))=x=g(x)^3+g(x)$$ As $$\ g(x) \rightarrow_{x\rightarrow \infty} \infty$$ ...
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1answer
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Approximate cos(58) to four decimal place accuracy using taylor's theorem

Approximate cos(58) to four decimal place accuracy using taylor's theorem.
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1answer
455 views

The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = ...
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3answers
228 views

How to prove $\int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}$, where $0 < c < 1$

This is an exercise from Apostol (p.285) that I'm having trouble with (in fact, I'm having trouble with the whole section): Prove that $\displaystyle{\int_0^1 \frac{1+x^{30}}{1+x^{60}} = 1 + ...
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3answers
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Bound for error term in Taylor expansion of $\arctan x$.

I'm trying to solve the following problem from Apostol, Calculus, Volume I (p. 284) and could use some help: Prove: $$\arctan x = \sum_{k=0}^{n-1} \dfrac{(-1)^k x^{2k+1}}{2k+1} + E_{2n} (x), ...
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1answer
212 views

lower bound for the radius of convergence of Taylor series [duplicate]

Possible Duplicate: Radius of convergence of power series In 4 years of studying physics I came across a lot of Taylor series. All of them converged in a disc with a radius equal to the ...
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1answer
384 views

Taylor Series Expansions

Problem with this question $$f(x)=3\sin x\cos \frac x 2 \text{ around } a=1$$ I have been breaking up $f(x)$ into $g(x) = 3\sin x$ and $h(x)= \cos \frac x 2$, and then multiplying them back ...
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1answer
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Clever derivation of $\arcsin(x)$ Taylor series

I was working the other day in the Math Help Centre, trying to help some first years with a calculus problem. The problem involved investigating the Taylor series of $\arcsin(x)$. Once the students ...
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1answer
176 views

Bernoulli and Euler numbers in some known series.

The series for some day to day functions such as $\tan z$ and $\cot z$ involve them. So does the series for $\dfrac{z}{e^z-1}$ and the Euler Maclaurin summation formula. How can it be analitically ...
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1answer
744 views

Taylor series expansion of $\sin(2x^2)$

I am doing homework an online calc class and having a tough time getting hold of the prof, so I was hoping someone here could let me know what I am doing wrong. I have to find the first few terms for ...
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1answer
111 views

Taylor Series and Size of Functions

I am looking for help with this question By looking at the Taylor series, decide which of the following functions $$\ln(1+y^2) \; \; \; \; \; \; \; \; \sin(y^2)\; \; \; \; \; \; \; \;1-\cos(y)$$ ...
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2answers
3k views

What is the Maclaurin series expansion for $\sqrt{x}$?

The derivative of $\sqrt{x}$ doesn't have a defined value at x = 0. How then do I find its maclaurin series expansion? Or can it only be approximated with a Taylor series at some value x != 0?
2
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1answer
594 views

Generalized binomial theorem

Prove that: $$(1+x)^{\alpha}=\sum_{n=0}^{+\infty}{\alpha \choose n} x^n$$ for $x\in[0;1), \alpha \in\mathbb{R}$ based on Taylor's theorem with Lagrange remainder. I don't feel such proofs. ...