let give a sequence of reals (a_n )_n / a_n >0 and
U_n = (a_n /((1+a_1 )(1+a_2 )....(1+a_n )))
1) prove that Σ u_n converges
2) calculate Σ u_n if u_n = (1/(√n)) .
let give I(x)= ∫_0 ^(π/2) (dt/(√(sin^2 t +x^2 cos^2 t))) and
J(x)= ∫_0 ^(π/2) ((cost)/(√(sin^2 t +x^2 cos^2 t)))dt cslculate lim_(x→0^+ ) (I(x)−J(x))
and prove that I(x)=_(x→0^+ ) −lnx +2ln2 +o(1).