# Tagged Questions

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.

75 views

### is it true that $\frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon}x$?

Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x)$ ($\epsilon, x \in (0,1)$). Here is one using $\ln (1+y) \approx y$: $$\frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x$$ I ...
49 views

35 views

### Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
61 views

### Find a bound for approximating Taylor Series

I'm struggling to figure out how to find a bound on my error for this problem: Let T_{6}(x) be the Taylor polynomial of degree 6 based at a = 0 for the function f(x)=\cos(x). Suppose you approximate ...
30 views

### Order Taylor series prediction

It is easy to add the Taylor expansion, less easy to multiply and even less easy to compose. That said, the main problem lies not in the calculation itself in the prediction orders that need to be ...
33 views

### Accuracy of Runge-Kutta compared to Taylor expansion

Let's say I have an ODE of the form : $y'(x) = f(x,y(x))$. I've been told that using the Runge-Kutta method for solving this ODE is equivalent to using Taylor expansion if $f(x,y(x))$ is linear in $y$ ...
13 views

### Sommerfield Expansion Taylor Expansion

Ugh... I can't figure this out and I DO NOT understand why. So this has to do with the Sommerfield expansion of the Fermi function (wiki) (another reference) The issue I'm having is we are supposed ...
75 views

### Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that…

Let $s(n,k)$ denote the signless Stirling numbers of the first kind. Prove that: $$s(n,2) = (n-1)!(1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n-1})$$ -I haven't dealt with Taylor series expansion ...
40 views

44 views

### Differentials squared - Divergence in general orthogonal curvilinear coordinates.

I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ...
30 views

### $\exists C>0$ such that $\frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2-Cx}$?

It is well-known and easy to check that for any real $x$ it holds $$\frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2}.$$ [To show this, it is sufficient to write explicitly their Taylor series ...
19 views

### Two dimensional taylor expansion of arbitrary function

Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
21 views

### Show a function behaves as a harmonic oscillator

We have a function $V(x)$ (potential energy) with $x$ being some variable. This function has a minimum at a certain $x_0$. We assume that $V(x)$ is an analytic real function of $x$ around $x_0$. ...
37 views

36 views

### Use Taylor Series to arrive at the expression for the forward approximation for a derivative [duplicate]

Use Taylor Series to arrive at the expression for the forward approximation for a derivative. $$f'(x)\approx \frac1h\left(-\frac32f(x)+2f(x+h)-\frac12f(x+2h)\right)$$ I'm not sure how to even go ...
98 views

33 views

### Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
31 views

### Finding a Maclaurin series

I have a question here; suppose $f(x)= x^2\sin(x^3)$ By using the Maclaurin series for sine, find the Maclaurin series for $f$ I understand how to obtain the Maclaurin series for $f$ using the ...