Find the coefficient of Taylor series
What is the coefficient of $x^n$ for the Taylor series $(1+x)^p$? What is the interval of convergence (R)?
I can write the Taylor series as- $$f(x) = 1+px+\frac{p(p-1)}{2!}x^2 +\frac{p(p-1)(p-2)}{3!}x^3 +... $$ But I can't find the general formula $(a_n)$ for the coefficient of $x^n$.
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$\begingroup$We recall the Taylor series expansion of $f$ around $x=0$ is given as \begin{align*} f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n\tag{1} \end{align*}
We assume $p$ is a non-negative integer. In this case the function $f(x)=(1+x)^p$ can be written according to the binomial theorem as \begin{align*} f(x)&=(1+x)^p=\sum_{n=0}^p\binom{p}{n}x^n\\ &=\sum_{n=0}^{p}\frac{p!}{(p-n)!n!}x^n\tag{2} \end{align*}
Comparison of (1) and (2) shows that (2) is already the Taylor expansion of $(1+x)^p$ around $x=0$ with coefficients \begin{align*} \frac{f^{(n)}(0)}{n!}=\begin{cases} \binom{p}{n}&0\leq n\leq p\\ 0&n>p \end{cases} \end{align*} Since $(1+x)^p$ is a polynomial of degree $p$ it converges for each $x\in\mathbb{R}$, so $R=\infty$.
Hint: The more general framework of binomial series expansion and especially the regions of convergence might be interesting.
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