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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Using Integrals to Derive the Taylor Series

An answerer gave a derivation (Where do the factorials come from in the taylor series?) for the standard form of the taylor polynomial series, copied and pasted below for ease of viewing. I wanted to ...
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Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na \...
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1answer
30 views

Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
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2answers
71 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
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1answer
32 views

Does $f(x) = ln(1+2x+2x^2) - 2x$ have a critical point at x = 0?

If we taylor expand $f(x)$ we get: $f(x) = \frac{-4}{3}x^3+O(x^4)$ We also know that $f(0) = 0$. The correct answer is no, because f(x) will be negative for positive x close to zero, and positive ...
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General question on Taylor Series

The Taylor Series comes from an assumption that a function has an expression as power series. Given such assumption we can then say that the $n$-th derivative and evaluate them at $x = a$, it can give ...
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Find Taylor series of function around $x=0$

I'm trying to calculate the Taylor serie around $x=0$ of the function $$f(x)=\int\limits_0^xe^{-t^2}dt$$ I tried to use the fundamental theorem of calculus, but I'm still stuck.
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Multiplicative version of Maclaurin or Talyor series

Is there a multiplicative version of Maclaurin or Talyor series? May be in the format $\ln y = b_0 + b_1 \ln x + b_2 (\ln x)^2 + \cdots $ I want to use that as an approximation in a regression ...
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4answers
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What does it mean that a taylor series generated for a function f(x) doesnt converge to f(x)?

If a some function f(x) is continous and has derivatives of all orders on some interval I, and assuming that f(x) can be expressed as a power series on I. And now you generate a taylor series for f(x),...
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How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
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114 views

Engineering Mathematics Problem with Taylor's Series

This is a problem from Engineering Mathematics book by K.A. Stroud 7th edition, Exercise 18, Chapter 12 Further problems. It has been given in a physics manner, but it just requires manipulation of ...
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1answer
28 views

Taylor expansion of an expectation

Ok guys, I'm reading a book and I'm not getting quite well a concept. If I have to expand $U'(Y_0(1+r_i))$ around $Y_0(1+r_f)$, why I get this: $\mathbb{E}[U'(Y_0(1+r_i))(r_i-r_f)]=U'[Y_0(1+r_f)]\...
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1answer
28 views

Taylor Series substitution giving different answers

I was given the function: $f(x) = 1/(1+x)^2$ and its Taylor series: $1 - 2x + 3x^2 - 4x^3 + \cdots$ In order to get the Taylor series for the closely related function $1/((1/2)+x)^2$, I simply ...
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1answer
28 views

Taylor's series and nth derivitive

The problem is: Calculate the Taylor's series in "$a=1$" of the function : $$f(x)=(5x-4)^{-\frac{7}{3}}\ .$$ I've started off by calculating the $n$th derivative of a function : \begin{align} f'(x)&...
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Bounding $f'$ in terms of $f$ and $f''$

Assume that $f: \mathbb{R} \to [0,\infty)$ is $C^2$ and $|f''(x)| \leq A$ for all $x$. Show that the inequality $$(f'(x))^2 \le 2Af(x)$$ holds for all $x$. The hint given in the question was, "...
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1answer
78 views

Is square root of Taylor series of $f(x)$ equivalent to the Taylor series of square root of $f(x)$

Mathematica treats two expressions as they are equivalent: Sqrt[Series[y[x], {x, x0, 1}]] Series[Sqrt[y[x]], {x, x0, 1}] Is that mathematically justified? Is ...
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237 views

How to find all roots of the quintic using the Bring radical

Finding one root $x_1$ of the quintic equation $x^5 + x = -a$ by using the Bring radical is described on Wikipedia. The root is $x_1 = -a +a^5 -5a^9+35a^{13}+ \ldots$ , and it is found by reversion ...
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Finding $f^{(2015)}(0,0,0)(x,y,z)^{2015}$ if $f=xe^{1-xy}+ \frac{z}{1-z^2}+z^{3}\sin(x+y).$

$$f^{(2015)}(0,0,0)(x,y,z)^{2015}$$ $$f=xe^{1-xy}+ \frac{z}{1-z^2}+z^{3}\sin(x+y).$$ I will give you my thoughts as soon as I type out an example from class that makes sense to me. Use of Taylor ...
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1answer
36 views

Understanding central difference formula for computing numerical gradient

More can be found here: http://www.math.ohiou.edu/courses/math3600/lecture27.pdf. I'm having trouble understanding what happens to the $h$ in this example where the central difference error is ...
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91 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
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1answer
47 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: $$x+r\frac{e}{e+t+x^2u}+\frac{xr^2\frac{e}{e+t+x^2u}}{1+r\frac{...
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1answer
96 views

If all derivatives are zero at a point, what does this imply?

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, ...
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2answers
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Calculating $f'(x)$ with $f(x)$ and a relative error?

I want to calculate $f'(x)$ using the formula: $$ f'(x) = \frac{f(x+h) - f(x)}{h}$$. Of course the error here is $o(h)$. However, what if in measuring $f(x)$ and $f(x+h)$ I have a relative error of $\...
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39 views

Interval of convergence for a power series with $x^{2n}$

By definition, the radius of convergence (which is equivalent to the interval) is: $$R:=\frac{1}{\varlimsup_{n\rightarrow+\infty}\sqrt[n]{|a_n|}}$$ Where $\varlimsup_{n\rightarrow+\infty}$ is the ...
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1answer
50 views

Trapezodial Rule Error Proof (taylor)

I search for a proof of the (local) error of trapezodial rule using taylor series. I can only find proofs for the error of the rectangle rule and for trapezodial it's always just "similar" whatever ...
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Expression for variance using Taylor series

I have the following expression for the variance: $$Var[\hat{f_n}(x)]=\frac{1}{2nh}\cdot\frac{(F(x+h)-F(x-h))}{2h}\cdot((1-(F(x+h)-F(x-h)))$$ If $h \downarrow 0$, this is supposed to be equal to: $$...
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Taylor expansion of $f(x,y)=xy-x+2x^3-yx^3$ about (0,1)…

I am asked to expand $f(x,y)=xy-x+2x^3-yx^3$ about (0,1) up to second order: First I found the required derivatives, and their values at (0,1), $ f_x=y-1+6x^2-3yx^2=0$ $f_y=x-x^3=0$ $f_{xx}=12x-...
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1answer
61 views

Taylor Expansion for a two-variable function

I am having a lot of difficulty understanding the given notations for Taylor Expansion for two variables, on a website they gave the expansion up to the second order: $f(x,y)+[f_x+f_y]+\frac{1}{2!}[...
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Error term Taylor expansion

We have $E[\hat{f_n}(x)]=\frac{F(x+h)-F(x-h)}{2h}$, $h\downarrow0$. In order to compute this expectation I need to use a Taylor expansion, under the assumption that f' and f'' exists: $F(x+h)=F(x)+hf(...
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2answers
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Proving that for any Differentiable distribution $F(x)$, an expression is increasing in $x$?

I am guessing that for a continuous random variable on $[0,1]$, $$ U(x)=\Big[x F(x) + \int_x^1 (1-t)f(t)dt\Big]x $$ is increasing for any distributions, because I can show $$ U'(x)=2xF+x^2f+\int_x^1 ...
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calculating the taylor series when there is an integral involved

one of the exercises is to calculate the taylor expansion at x=0 and degree 4 for some function. For example: $$\int_{0}^{x} e^{-t^{2}} dt$$ I actually have no clue how to get started. I know how to ...
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1answer
50 views

Is there any standard method for finding the function defined by a Taylor/Laurent series?

Say you have a Taylor series defined by $$\sum_{n=0}^{\infty}a_nx^n$$ Is there any standard way to figure out what function is defined by the series? One option I see is just looking at the ...
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62 views

Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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Problem with Maclaurin series expansion method.

Look at the following series: 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ..... You can say by using any method that the series is divergent. It indeed diverges but we use this as a series expansion for 1/(1-x)^2. ...
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Taylor Series of $\sin x/(1-x)$

Ιs there any fast way to calculate the first four non-zero terms a Taylor Series $\dfrac {\sin x}{1-x}$ at $x=0$ without making big derivatives calculations? I know that $$\sin x = x- \frac{x^3}{6} + ...
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1answer
38 views

Lévy-Khintchine formula and Taylor expansion

I have trouble finding out why this condition $\int_{\mathbb{R}\backslash\{0\}} \min(1, x^2 ) \nu(dx) < \infty$ in the Lévy-Khintchine formula is necessary. The Lévy-Khintchine formula is ...
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Reverse Taylor series for sine

I want a little help with reverse Taylor series for sinus if is possible :D .From what I read the formula is: RadOfAngle - RadOfAngle^3*3! + RadOfAngle^5*5! - RadOfAngle^7*7! = Sins value. How can I ...
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How to show $K = O(\frac{\log x}{\log\log x})$ in this case?

How to show $K = O(\frac{\log x}{\log\log x})$ when $K$ is the smallest number for the following inequality to hold: $$ \sum_{k=K+1}^\infty \frac{(\ln2)^{k-1}}{k!} \leq \frac{1}{x} $$ This observation ...
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Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
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Question involving Taylor series and continuity

Question: $$f(x)=\lim_{n\rightarrow \infty}\frac{x^{2n}-1}{x^{2n}+1}$$ Where is this function continuous? Trial: I analyzed positive terms of x.For large values of n the function approaches to ...
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How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
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Looking for a way to apply the Taylor Series expansion to find derivatives for a function.

This post references the Riemann-Siegel formula found at here and at here. I am writing a Java program which implements this formula. I am having trouble with the remainder terms. The Riemann-Siegel ...
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67 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of $\...
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How to expand the $\ln(x)$ to Maclaurin series?

There was a silly question - how to expand the $\ln{x}$ to Maclaurin series?
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44 views

Logarithms and Taylor Series

Before Log Tables, how were they able to compute expressions such as $2^{2.221}$? I understand they could take a Taylor expansion of $\frac{1}{x}$, but how were they able to condense the expansion ...
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General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...
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78 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
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292 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
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79 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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5answers
172 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...