| Exercice 1. Soit {f\in{\mathcal C}^{0}([0,1],\mathbb{R})}. On pose {I_{n}=\displaystyle\displaystyle\int_{0}^{1}t^{n}f(t)\,\text{d}t} et {J_{n}=\displaystyle\displaystyle\int_{0}^{1}t^{n}\,\text{d}t}. Déterminer la limite de la suite {n\mapsto I_{n}\,⁄\,J_{n}}. |
| Exercice 2. Montrer que {\displaystyle\lim_{n\rightarrow+\infty}\displaystyle\int_{0}^{\pi/2}\cos^{n}(t)\text{d}t=0}. |
| Exercice 3. Montrer que {\displaystyle\lim_{n\rightarrow+\infty}\displaystyle\int_{0}^{+\infty}\dfrac{\text{d}t}{t^{n}+\text{e}^{t}}\text{d}x=1-\dfrac{1}{\text{e}}}. |
| Exercice 4. Montrer que {\displaystyle\lim_{n\rightarrow+\infty}\displaystyle\int_{0}^{+\infty}\!\!\!\dfrac{t^{n}}{1+t^{n+2}}\,\text{d}t=1}. |
| Exercice 5. Montrer que {\displaystyle\lim_{n\rightarrow+\infty}\displaystyle\int_{-\infty}^{+\infty}\Bigl(1+\dfrac{t^{2}}{n}\Bigr)^{-n}\text{d}t=\sqrt{\pi}} |
| Exercice 6. Soit {u_{n}=\!\!\displaystyle\int_{0}^{1}\!\!\dfrac{\text{d}t}{{(1+t^{3})}^{n}}}. Montrer {\displaystyle\lim_{+\infty}u_{n}=0}. Préciser la nature de la série {\displaystyle\sum u_{n}}. |