# 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.

28 views

### 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 ...
31 views

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 ...
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)&...
139 views

### 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, "...
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 ...
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 ...
68 views

### 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 ...
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 ...
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 ...
47 views

99 views

20 views

67 views

### How to expand the $\ln(x)$ to Maclaurin series?

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

### 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 ...
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 ...
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 ...
### 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 $f(x)<f(0)$, ...