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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2
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48 views

Why is this true? $\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty) : a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$

I know $$\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty)$$ $$a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$$ can be proved using AM-GM. Is there a simple way to show the inequality holds in all ...
2
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2answers
31 views

Approximation of a ratio

Is this approximation true? If so, why? $$\frac{1+x}{1+y}\approx 1+x -y$$ I think it has something to do with $x$ and $y$ being close to zero, so that the ratio of the two is approximately equal to ...
1
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1answer
32 views

Laplace equation, Taylor expansion

I couldn't find it anywhere, so I decided to write my question here: I have problems solving this equation: $$u_{xx} + u_{yy} = 4,$$ subjected to the conditions $$u(x,x)=2x^2, \quad u_x(x,x)=2x$$ ...
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0answers
21 views

Method to linearize a function?

I have a function $$ g(x) = x^{\frac{\beta}{x+x_o}} $$ where $\beta$ and $x_o$ are constants. I follow the usual steps and expand up to the first order around point $a$: $$ g(x) \approx g(a) + ...
0
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1answer
39 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
1
vote
1answer
46 views

Proof that sum of power series equals exponential function?

I have found that the Sum series equal an exponential function as below, however I have not found a proof for it: $$ ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!} $$ I have though managed to prove ...
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0answers
12 views

Symmetry of convergence around a of a Taylor Polynomial

I heard the lecturer mention this shortly, so I'd like a more detailed explanation of it. He showed us a taylor approx. of ln(x) at x = 1, and showed that it only matches in the interval ]0,2], so ...
0
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0answers
9 views

Estimating error for a general n? - Taylor

So, I think I kind of understand the idea of estimating the error $R_n f$ when n is given. We use that it must be equal to $\frac{f^{n+1}(z)}{(n+1)!} \cdot (x-a)^{n+1}$. We calculate $f^{n+1}(z)$ and ...
0
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2answers
19 views

Taylor series of polynomial.

I know that the taylor approx. of a polynomial centered at 0, if n gets big enough, is just the polynomial itself. But why do people always say "centered at 0"... wouldn't we also get the polynomial ...
0
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0answers
24 views

Finding an optimal path for minimizing an integral.

Let $x,y$ be real numbers. Let the function $f(x,y)$ be real-entire in both $x$ and $y$. Thus $f(x,y)$ is a real-entire Taylor series in the variables $x,y$. How the find a non-intersecting path ...
0
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1answer
13 views

Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
0
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1answer
11 views

Bounding $|n^2(1-(\cos\frac{1}{n})^2)|$

I'm working on a a problem that involves me needing to give an upper bound for the following expression: $|n^2(1-(\cos\frac{1}{n})^2)|$ My attempts at bounding it: Expanding the expression: ...
0
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1answer
43 views

On the multivariable Taylor expansion

Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ ...
1
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2answers
58 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
0
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2answers
29 views

Proof of an inequality using Newton's Method

Question: Show that the function $f(x):= x^3 -2x -5$ has a zero $r$ in the interval $I:= [2,2.2]$. If $x_1 :=2$ and if we define the sequence $(x_n)$ using Newton's procedure, show that $|x_{n+1} -r| ...
1
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1answer
58 views

Is there a difference between $a \cdot a^T$ and $a^2$?

The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a ...
1
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0answers
28 views

Functions f(x) equal to Taylor series vs Fourier series vs Bessel series

(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an ...
0
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3answers
53 views

If subsequent terms keep getting larger, does that mean no limit exists?

Take the following Taylor expansion: $$ \dfrac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots $$ This only holds for $ 0 \leq x < 1. $ Let's say you want to prove this doesn't hold for $x>1$. You can ...
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1answer
29 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...
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0answers
18 views

Taylor expansion of a scalar function

I have an expression on the form $$ \sum_{i=1}^N{\rho_i}f(\mathbf x+\mathbf c_i)\mathbf c_i $$ where $\rho_i$ is a scalar, $f(\mathbf x+\mathbf c_i)$ a scalar function of the vector quantities ...
0
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1answer
32 views

Real Analysis: Taylor's Theorem Approximation Proof

If $x>0$ Show that $\lvert (1+x)^{(1/3)} - (1+\frac{x}{3} -\frac{x^2}{9}) \rvert \le (\frac {5}{81})x^3$. Use this inequality to approximate $1.2^{1/3}$ & $2^{1/3}$. That is the actual ...
3
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1answer
54 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
0
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1answer
61 views

Constructing a sequence of function with bounded derivative

Let $f:\mathbb R\mapsto\mathbb R$ be a smooth function and analytic at $x=0$. I wish to find a sequence of functions $\{f_n\}$ such that $\{f_n(x)\}$ is convergent to $f(x)$ for all $x$ and $f'''_n$ ...
6
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0answers
77 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
1
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2answers
26 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
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6answers
39 views

Taylor expansion square

Consider the following expansion $$\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \dfrac{1}{16}x^3 .. $$ Show this equation holds by squaring both sides and comparing terms up to $x^3$. I ...
0
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1answer
33 views

Prove $\frac{dy}{dx}$ is approximated by $\frac{y(x+h)-y(x-h)}{2h}$ to $O(h^2)$

I tried to solve it by truncating the Taylor series expansions for $y(x+h)$ and $y(x-h)$ but I couldn't find a way to relate it to the derivative. I wasn't sure where the appropriate place to truncate ...
0
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2answers
42 views

Why cant we do substitution in differentiation but is ok in taylor series?

I have the same question 10 year ago when i was studying high school. I dont understand it and i give up the math. 10 year ago, i need to work with calculus during work and this question come to find ...
2
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2answers
47 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
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1answer
18 views

error order evaluation in taylor expansion of a definite integral

I have a function $g(x)=f(x)e^{-x}$ and i want to consider the following integral: $\int_{0}^{\infty}g(x)dx$. Since $f(x)$ is a complicated, but monotonic decreasing, function in the interval ...
0
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1answer
28 views

Prove error bound using Taylor's series Error term (Bound doesn't seem to make sense)

I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is ...
0
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1answer
25 views

I do not know the point at which this Taylor series was derived, can someone explain please?

I am required to derive Euler's method through Taylor's Theorem. I have been given the Taylor series for $y(t)$ as shown below. However I do not understand what point the Taylor series was derived. ...
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1answer
14 views

Error estimate of definite integral of a taylor expanded function

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute ...
3
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2answers
34 views

Other ways to evaluate $\lim_{x \to 0} \frac 1x \left [ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1\right ]$?

Using the facts that: $$\begin{align} \sqrt{1 + x} &= 1 + x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt{1 - x} &= 1 - x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt[3]{1 + x} &= 1 + x/3 + \mathcal{o}(x) ...
0
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1answer
27 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...
0
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0answers
31 views

Finding a limit using Taylor's theorem

let's say that g(x,y) is $c^{n+1}$ and let's say that p(x,y) is it's n-th order Taylor polynomial. I am trying to prove that: $$\lim_{(x,y)\to (0,0)} \frac{g(x,y)-p(x,y)}{(\sqrt{x^2+y^2})^n}=0$$ I ...
0
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3answers
54 views

(Taylor's theorem) Proving that $\sin(x) = \sum\limits_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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1answer
40 views

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$.

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$. What is its radius of convergence? So I write the fraction as $\frac{1}{(x-1)(x+3)}$ and what should I do now?
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0answers
15 views

cos(x) approximation with taylor of second degree

there is an approximation to find cos(x) is 1 - (x^2)/2, until n = 2 degree of taylor, but I'm confuse how to find how good is its approximation, the one thing I know only I get its error is (sin(c) ...
2
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1answer
54 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
-1
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1answer
53 views

How to derive Maclaurin series for ln(1+x) without calculus?

How can we show $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$ for $-1 < x \leq 1$ without using calculus?
6
votes
0answers
88 views

Taylor expansion of $x^{1/x}$

I am new to Taylor expansions and I would like to calculate the Taylor polynomial of the function $x^{1/x}=e^{(1/x)\log x}$. Since the function is not defined at $x=0$, how should I choose the point ...
2
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3answers
66 views

Does the Taylor series of an infinitely differentiable function converge; and if yes, does it converge to the function? [duplicate]

I have googled it, but I am not satisfied with those. So my questions are: Let $D$ be an open set in $\mathbb{R}$. Let $f:D\rightarrow \mathbb{R}$ be a infinitely differentiable ...
0
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1answer
16 views

Convergence of Taylor series about centre of open disc for analytic function.

I define a function on an open set of the complex plane to be analytic if about any point $z_0$ in that set it can be expanded as a power series in $(z - z_0)$ that converges in some neighbourhood of ...
0
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0answers
6 views

Taylor Polynomial to estimate solution of MVC differential equation

I have read a few examples, that you're using derivatives at different points to estimate a polynomial but I need a bit a of guidance to understand how this would work in a multivariate calculus ...
0
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1answer
40 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...
0
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1answer
54 views

Sum of Taylor Series

I have the converging series: $$ 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... $$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
0
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1answer
30 views

Infinite series expansion of $\arcsin (x)$ and $\arccos (x)$

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?
0
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0answers
24 views

Taylor Series Expansion

PROBLEMS ax^2 + x + 1 = 0 (1) 1. Using a Taylor series expansion express the solution to the quadratic equation in Equation (1) as a series. Include terms up to cubic order. Find the cubic term in ...
0
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0answers
15 views

Is there an expression for $\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = $?

Does an expression for $$ \exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ? $$ exist? For j=1 we have the usual expression for translation and scaling $$ \exp\left( t \partial_z\right) f(z) = ...