grandes-ecoles 2023 Q10

grandes-ecoles · France · centrale-maths1__official Sequences and Series Properties and Manipulation of Power Series or Formal Series

Let $\left(a_k\right)_{k \in \mathbb{N}}$ and $\left(b_k\right)_{k \in \mathbb{N}}$ be sequences of elements of $\mathbb{K}$ such that $\sum_{k=0}^{+\infty} a_k D^k = \sum_{k=0}^{+\infty} b_k D^k$.
Show that, for all $k \in \mathbb{N}$, $a_k = b_k$.

Let $\left(a_k\right)_{k \in \mathbb{N}}$ and $\left(b_k\right)_{k \in \mathbb{N}}$ be sequences of elements of $\mathbb{K}$ such that $\sum_{k=0}^{+\infty} a_k D^k = \sum_{k=0}^{+\infty} b_k D^k$.

Show that, for all $k \in \mathbb{N}$, $a_k = b_k$.

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