grandes-ecoles 2019 Q31

grandes-ecoles · France · centrale-maths1__psi Sequences and Series Power Series Expansion and Radius of Convergence

In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$, $$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$ Fix an integer $n \geqslant 2$.
Using the result of question 30 and by expanding $(1-t)^{n+1}$, determine the Taylor expansion of $g_{n}$ to order $n$ at 0.

In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$,
$$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$
Fix an integer $n \geqslant 2$.

Using the result of question 30 and by expanding $(1-t)^{n+1}$, determine the Taylor expansion of $g_{n}$ to order $n$ at 0.

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