Bonsoir,
On suppose vrai [tex]U_n \leq 2-\dfrac{1}{n}[/tex]
On va montrer que [tex]U_{n+1} \leq 2-\dfrac{1}{n+1}[/tex]
[tex]n \ \textgreater \ 0 \\
\Longrightarrow \dfrac{1}{n*(n+1)^2}\ \textgreater \ 0\\
\Longrightarrow \dfrac{1}{n }-\dfrac{1}{(n+1)^2} -\dfrac{1}{n+1}\ \textgreater \ 0\\
\Longrightarrow \dfrac{1}{n }-\dfrac{1}{(n+1)^2} \ \textgreater \ \dfrac{1}{n+1}\\
\Longrightarrow -\dfrac{1}{n }+\dfrac{1}{(n+1)^2} \ \textless \ -\dfrac{1}{n+1}\\
\Longrightarrow 2-\dfrac{1}{n }+\dfrac{1}{(n+1)^2} \ \textless \ 2-\dfrac{1}{n+1}\\
\Longrightarrow U_{n+1}=U_{n}+\dfrac{1}{(n+1)^2} \ \textless \ 2-\dfrac{1}{n }+\dfrac{1}{(n+1)^2} \ \textless \ 2-\dfrac{1}{n+1}\\
\Longrightarrow U_{n+1}\ \textless \ 2-\dfrac{1}{n+1}\\
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