Exo1: Let the series \( \sum h_{n} \) such as, \( \forall n \in \mathbb{N}, \forall x \in \mathbb{R}_{+} ; h_{n}(x)=(-1)^{n} \frac{n^{2}+x}{1} \) 1. Prove that que \( \sum h_{n} \) do not absolutely convergent at any point of \( \mathrm{R}+ \). 2. Prove that que \( \sum h_{n} \) converges uniformly on \( \mathrm{R}_{+} \) Voir : Asanalyse 1-2-3 Guelma.pdf exo1 page 372

To prove that the series \( \sum h_{n} \) does not absolutely converge at any point of \( \mathbb{R}_{+} \), we need to show that the series of the absolute values of the terms of the series does not converge. Given that \( h_{n}(x) = (-1)^{n} \frac{n^{2}+x}{1} \), we can find the absolute value of \( h_{n}(x) \) as \( |h_{n}(x)| = \left| (-1)^{n} \frac{n^{2}+x}{1} \right| = \frac{n^{2}+x}{1} \). Now, let's consider the series of the absolute values of the terms of the series: \[ \sum_{n=1}^{\infty} |h_{n}(x)| = \sum_{n=1}^{\infty} \frac{n^{2}+x}{1} \] To prove that this series does not converge, we can use the comparison test. We will compare this series with the series \( \sum_{n=1}^{\infty} n^{2} \), which is a divergent series. Since \( n^{2} \) grows faster than \( n^{2}+x \) for any fixed \( x \in \mathbb{R}_{+} \), the series \( \sum_{n=1}^{\infty} n^{2} \) diverges. Therefore, by the comparison test, the series \( \sum_{n=1}^{\infty} |h_{n}(x)| \) also diverges. Hence, the series \( \sum h_{n} \) does not absolutely converge at any point of \( \mathbb{R}_{+} \). Next, to prove that the series \( \sum h_{n} \) converges uniformly on \( \mathbb{R}_{+} \), we need to show that the limit of the partial sums of the series converges uniformly for all \( x \in \mathbb{R}_{+} \). Let's consider the partial sums of the series: \[ S_{n}(x) = \sum_{k=1}^{n} h_{k}(x) = \sum_{k=1}^{n} (-1)^{k} \frac{k^{2}+x}{1} \] To prove that the series converges uniformly, we need to show that the limit of the partial sums converges uniformly for all \( x \in \mathbb{R}_{+} \). Since the series \( \sum_{k=1}^{\infty} (-1)^{k} \frac{k^{2}+x}{1} \) converges, the limit of the partial sums converges uniformly for all \( x \in \mathbb{R}_{+} \). Therefore, the series \( \sum h_{n} \) converges uniformly on \( \mathbb{R}_{+} \).