grandes-ecoles 2019 Q24

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Limit Evaluation Involving Sequences

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

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

Paper Questions

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