Bonsoir,
1)
[tex]u_n=\dfrac{1}{n(n+1)}\\\\u_{n+1}-u_n=\dfrac{1}{(n+1)(n+2)}-\dfrac{1}{n(n+1)}\\\\=\dfrac{n-(n+2)}{n(n+1)(n+2)}\\\\=-\dfrac{2}{n(n+1)(n+2)}\\\\[/tex]
[tex](u_{n}) \textrm{ est d\'ecroissante.}[/tex]
2)
a)
[tex]\dfrac{1}{n} -\dfrac{1}{n+1}=\dfrac{(n+1)-n}{n(n+1)}=\dfrac{1}{n(n+1)}\\[/tex]
b)
[tex]u_1=\dfrac{1}{1(1+1)}=\dfrac{1}{1*2}=\dfrac{1}{1}-\dfrac{1}{2}\\\\u_2=\dfrac{1}{2(2+1)}=\dfrac{1}{2*3}=\dfrac{1}{2}-\dfrac{1}{3}\\\\u_3=\dfrac{1}{3(3+1)}=\dfrac{1}{3*4}=\dfrac{1}{3}-\dfrac{1}{4}\\\\ ...\\\\u_{p-1}=\dfrac{1}{p-1}-\dfrac{1}{p}\\\\u_{p}=\dfrac{1}{p}-\dfrac{1}{p+1}\\\\\textrm{on additionne membre \`a membre et on simplifie} S_p=1-\dfrac{1}{p+1}\\\\[/tex]
c)
[tex]=1-\dfrac{1}{1000} =0.999[/tex]
