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 […]

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, […]

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..

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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!} […]

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