grandes-ecoles 2018 Q19

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Power Series Expansion and Radius of Convergence

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q18, deduce that $f$ is expandable as a power series on $]-1,1[$ and that $$\forall x \in {]-1,1[}, \quad f(x) = \sum_{k=1}^{+\infty} (-1)^{k} \zeta(k+1) x^{k}$$

Let $f$ be the function defined by
$$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$
Using the result of Q18, deduce that $f$ is expandable as a power series on $]-1,1[$ and that
$$\forall x \in {]-1,1[}, \quad f(x) = \sum_{k=1}^{+\infty} (-1)^{k} \zeta(k+1) x^{k}$$

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