(211)
Find the derivative of Δx, where
Δx= determinant (((f_1 (x)),(φ_1 (x)),(Ψ_1 (x))),((f_2 (x)),(φ_2 (x)),(Ψ_2 (x))),((f_3 (x)),(φ_3 (x)),(Ψ_3 (x))))
and f_1 (x) ,f_2 (x), f_3 (x),φ_1 (x), etc. are different functions of x.
If f(x,y)=(((x^2 +y^2 )^n )/(2n(2n−1)))+xφ((y/x))+Ψ((y/x)),
then using Euler′s theorem on homogenous functions,show that
x^2 ((δ^2 f)/(δx^2 ))+2xy((δ^2 f)/(δxδy))+y^2 ((δ^2 f)/(δy^2 ))=(x^2 +y^2 )^n
u_n = Σ_(k=n+1) ^(2n) (1/k) and v_n = Σ_(k=n) ^(2n−1) (1/k)
• show that u_n and v_n are adjacent
use ln(x+1) ≤ x and x≤−ln(1−x) and
• show that u_n ≤ Σ_(k=n+1) ^(2n) (ln(k)−ln(k−1))
hence deduce that u_n ≤ ln2
• show that v_n ≥ Σ_(k=n) ^(2n−1) (ln(k+1)−ln(k))
hence deduce that v_n ≥ln2