# 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

51 views

### Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
161 views

47 views

### Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...
32 views

### zero of second order

I'm studying functions associated with a domain in the complex plane. In one paper that I'm reading, a particular function, $R(a, b)$, is discussed (with "$a$" varying and "$b$" fixed complex ...
35 views

### Any way to characterize this family of polynomials?

I have a family of polynomials generated by the recurrence relation $P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$ The family is related to the Lambert $W$-function by its ...
35 views

### A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
13 views

### Taylor Series function seperation.

Say we have a function $$F= \frac{g}{h}$$ And we want to expand it with Taylor series keeping only second grade terms. How do we know when to expand $F$ directly or $g$ and $\frac{1}{h}$ ...
11 views

### Taylor series of complex function

$f(z) = \frac{2}{z^2-1}$ at $z = i$ My solution: $t = z - i$ $z = t + i$ $\frac{2}{z^2-1} = -2\frac{1}{1-(t+i)^2} = -2\sum_{n=0}^\infty (t+i)^{2n} = -2\sum_{n=0}^\infty z^{2n}$ Where is my ...
50 views

43 views

### Calculus 2 - Prove Disprove - convergence of Taylor series

I got this question regarding properties of Taylor series. I'm stuck on the second question, I believe it is true since the area of convergence for X is affected by the coefficient and it is not ...
15 views

### About the set of $x$ values at which the Taylor series of $f(x)$ converges to $f(x)$

Let $f(x)$ be a function (for simplicity, let us assume that it is defined on $\mathbb{R}$ and infinitely differentiable), and $T$ the Taylor series of $f$ at $x=a$, with interval of convergence $I$. ...