grandes-ecoles 2019 Q23

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Convergence/Divergence Determination of Numerical Series

For every natural integer $n$ and every real $x$ in $J = [0, 1/2[$, set $$S_n(x) = \sum_{p=1}^{+\infty} \left(\sum_{k=n+1}^{+\infty} \frac{2^{2p+1} x^{2p-1}}{(2k-1)^{2p}}\right).$$ Justify that, for every natural integer $n$, the function $S_n$ is defined on $J$.

For every natural integer $n$ and every real $x$ in $J = [0, 1/2[$, set
$$S_n(x) = \sum_{p=1}^{+\infty} \left(\sum_{k=n+1}^{+\infty} \frac{2^{2p+1} x^{2p-1}}{(2k-1)^{2p}}\right).$$
Justify that, for every natural integer $n$, the function $S_n$ is defined on $J$.

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