# Questions tagged [taylor-expansion]

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.

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### 'deducing' a bound using the first order taylor series. How to make it more precise?

So, I just saw a ‘proof’ that the generalized birthday problem has a median of C*sqrt(n). Though the probability in question is interesting, this question is more about calculus and maybe asymptotics ...
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### Construct/prove existence of a function with given expansions at two different points

Consider two non-constant real polynomials $f(x)$ and $g(x)$: $$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N$$ $$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M$$ where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ ...
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### How to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$

In the calculation of a problem, I need to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$ holds for any positive integer $n$. I got the expansion ...
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