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.

learn more… | top users | synonyms (1)

8
votes
0answers
152 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of ...
6
votes
0answers
86 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 ...
6
votes
0answers
59 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
4
votes
0answers
55 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
43 views

Am I missing anything when doing this taylor expansion?

I'll make my question short, I am encountering a error when doing expansion. I am expanding $f(x)=2x^3+4x+1$ and after the expansion, things don't match. Here's what I'm doing. Let $a=5$ ...
4
votes
0answers
357 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 ...
3
votes
0answers
27 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
3
votes
0answers
65 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
3
votes
0answers
123 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
3
votes
0answers
80 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 $$ ...
3
votes
0answers
311 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 $$ ...
3
votes
0answers
134 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of signi´Čücance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
3
votes
0answers
53 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
0answers
45 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
3
votes
0answers
74 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
3
votes
0answers
215 views

Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...
3
votes
0answers
66 views

Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for ...
3
votes
0answers
259 views

domain of convergence of a multivariable taylor series

consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
2
votes
0answers
66 views

How to express $e^{yS^2}(f(x))$ in closed form where $\frac{d}{dx}=S$

$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$ is a operator $$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
2
votes
0answers
38 views

Taylor polynomial, Peano form of the remainder of f(x) and its asymptote

Could someone help me complete this or check if my reasoning so far is correct? I'm stuck at finding the oblique asymptote: Write the Taylor polynomial and the Peano form of the remainder of $f(x) = ...
2
votes
0answers
28 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
2
votes
0answers
18 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
2
votes
0answers
27 views

Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
2
votes
0answers
36 views

Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
2
votes
0answers
40 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 ...
2
votes
0answers
214 views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
2
votes
0answers
79 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
0answers
68 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 $\frac{1}{x}$: $$ ...
2
votes
0answers
302 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
2
votes
0answers
206 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
2
votes
0answers
32 views

Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
2
votes
0answers
159 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$ Obviously, integer derivatives can be defined as $\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty ...
2
votes
0answers
55 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
2
votes
0answers
67 views

Taylor expansion proof

It's pretty clear to me that in this expansion: $$p(x) = f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+f'''(0)\frac{x^3}{3!}+\cdots$$ When I assume $x=0$, $p(0)$ is gonna be equals to $f(0)$ and all its ...
2
votes
0answers
179 views

Proving Lagrange's Remainder of the Taylor Series

My text, as many others, asserts that the proof of Lagrange's remainder is similar to that of the Mean-Value Theorem. To prove the Mean-Vale Theorem, suppose that f is differentiable over $(a, b)$ ...
2
votes
0answers
59 views

Taylor polynomial of $\frac{1}{1-x-y}$

I need to calculate the 2nd order Taylor polynomial at the origin of $$f(x,y) = {1 \over{1-x-y}}$$ I have looked at two ways, and not sure which is simpler. We can split it by partial derivatives ...
2
votes
0answers
134 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
2
votes
0answers
79 views

Determine the series for cos x^2

Use the series for Cos x (Taylor Series) If you could give me help or the solution to the problem, that would be great! Thanks
2
votes
0answers
53 views

Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
2
votes
0answers
64 views

How might I find all functions that have a power series whose coefficients are all 1 or -1.

Suppose I have that $f(x)= \pm 1 \pm x \pm x^2 \cdots$ How can I find $f(x)$? Specifically, I know I'm looking for all functions such that $|f^{k}(0)|=k!$, which functions are those? Which ...
2
votes
0answers
2k views

power series of arcsin(x) centered at x = 0

I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$. Sorry about the notation, I'm not sure what syntax to use. S stands ...
2
votes
0answers
311 views

Consistency order of backward Euler method

How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$ \vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1) $$ has a unique ...
2
votes
0answers
85 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
2
votes
0answers
62 views

Error while inverting a function using n terms of its Taylor series

I have $b>a$ and an invertible and infinitely differentiable function $f$. If I want to evaluate $\epsilon=f^{-1}(b-a)$ by writing: $b\approx ...
2
votes
0answers
113 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
2
votes
0answers
140 views

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 ...
2
votes
0answers
207 views

Asymptotic Series of Confluent Hypergeometric Function

I am using a generating function method to try and solve a recurrence. I have solved the resulting differential equation to find the generating function takes the form: $$A(z) = \frac{\, ...
2
votes
0answers
1k views

Taylor series and little-oh notation

Consider the series $e^{\tan(x)} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{3x^{3}}{3!} + \dfrac{9x^{4}}{4!} + \ldots $ Retaining three terms in the series, estimate the remaining series using ...
2
votes
0answers
207 views

4 vector Taylor expansion, sign confusion

I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from. $$ e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = ...