Question

Exercitiul 15 va rog mult de tot​

Answer 1

1/(xk-xk²) = -1/(xk²-xk) =

= -1/(xk(xk-1)) = -(1/(xk-1) - 1/xk) =

= 1/xk - 1/(xk-1)

=> suma 1/(xk-xk²) = 1/x1+1/x2+...+1/x20 -(1/(x1-1)+1/(x2-1)+...+1/(x20-1) =

= S19 / S20 + suma (1/(1-xk)) =

= 0 + suma (1/(1-xk)).

f'(x) = suma f(x)/(x-xk)

f'(1) = suma f(1)/(1-xk)

f'(1) = 5 suma 1/(1-xk)

(20+10+5) = 5 suma 1/(1-xk)

=> 35 = 5 suma (1/(1-xk)

=> suma (1/(1-xk)) = 7

$\Rightarrow \sum\limits_{k=1}^{20} \dfrac{1}{x_k - x_k^2} = 7$