Articles of taylor expansion

Taylor Polynomials, Why only Integer Powers?

So It seems that the definition of polynomial is that is is raised to an integer power, but why is this necessary? My question mainly arises from a proof of the solution to the Hydrogen atom in quantum mechanics, where they rely on the fact that the terminating term in series expansion of a function […]

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the following conclusion: $$(\sum_{n=0}^\infty c_nx^n)^p = \sum_{{n_1}=0}^\infty \sum_{{n_2}=0}^\infty \cdots \sum_{{n_p}=0}^\infty c_{n_1}x^{n_1} c_{n_2}x^{n_2} \cdots c_{n_p}x^{n_p}.$$ Now, for multiplying two different Taylor series, […]

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as this seems hard..). I know that \begin{equation*} (1+x)^{1/x}=e^{ln(1+x)/x}, \end{equation*} and the Maclaurin series for $e^x$ is easy to prove, so I think it’s a good direction..

Create function F() from Points

I would like to recreate a function only by knowing points on the graph. So I would have the points A(x/y) B(x/y) C(x/y) and would like to create its f() Is this possible? I heard this should be possible with a Taylor Series but to do a Taylor Series wouldn’t I need a f() and […]

Expanding $(1 – x + 2y)^3$ in powers of $x-1$ and $y-2$ with a Taylor series

I would like to do this. I observe that I can write $$f(x,y) = (1 – x + 2y)^3 = (2(y-2) – (x-1) + 4)^3.$$ It’s easy to do this via algebra directly. However, I’m asked to do it by computing the Taylor series. Is it correct to say that I could expand the function […]

Raise a power series to a fractional exponent?

In showing that $\log^\alpha{(1+x)}$ is $O((x)^\alpha)$ at $1$, for $\alpha>0$, one can note that $$\left ( \frac{\log{(1+x)}}{x} \right )^\alpha \overset{x\to 0}{\longrightarrow} \left ( 1\right )^\alpha = 1.$$ So we know that $$\log^\alpha{(1+x)}= (x)^\alpha + o((x)^\alpha).$$ But how would I find $\beta > \alpha$ and $c\in \mathbb{R}$ such that $$\log^\alpha{(1+x)}= (x)^\alpha + c(x)^\beta + o((x)^\beta)?$$ I’d […]

Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to order 2 terms using binomial series

I can expand $(x+2)^{1/2}$ by taking $y=x+1$ and using the binomial series to find a power series representation of $f$: $$\sqrt{y+1}=\sum_{k=0}^{\infty}{1/2\choose k}y^k=1+\frac{y}{2}+\frac{\frac{1}{2}\frac{-1}{2}y^2}{2!}+O(y^3)=1+\frac{x+1}{2}-\frac{1}{8}(x+1)^2+O(x^3) $$ And grouping terms $$\sqrt{x+2}=(1+1/2-1/8)+\frac{x}{4}-\frac{x^2}{8}+O(x^3)=\frac{11}{8}+\frac{x}{4}-\frac{x^2}{8}+O(x^3)$$ I am not sure how this is supposed to bring me closer to the taylor series representation, and in particular the expansion about $x=2$. Sorry if some of […]

Logarithmic terms in series expansions

Lately, I am encountering more and more example of series expansions that do include logarithmic terms. For example (section 1 (page 3-5) in hep-th/0002230) when taking a scalar field on an asymptotic AdS background in Fefferman-Graham coordinates $$ds^2 = \frac{l^2}{r^2}(dr^2 + G_{ij}(x, r) dx^i dx^j) $$ One can write a general asymptotic AdS spacetime in […]

A problem related to mean value theorem and taylor's formula

I guess I need to use Taylor’s formula and the mean value theorem. I have no idea except for them. Note: honestly, this is not homework. I am studying by myself. Suppose that $f\colon\mathbf{R}^2\to\mathbf{R}$ is $\mathcal{C}^p$ on $B_r(x_0,y_0)$ for some $r>0$. Prove that, given $(x,y)\in B_r(x_0,y_0)$, there is a point $(c,d)$ on the line segment […]

Taylor expansion of an stirling identity

I have been searching many ways for a week just to solve this, to no avail. I’m still confused about how the Taylor expansion is produced. It is so advanced compared to the subjects that I took. I am currently taking advance researches or work/journals from other mathematicians but I still cannot do this: $$\frac{(e^{w}-1)^{k}}{k!} […]