Question

Вычислите сумму 1/2x4+1/4x6+...1/2n(2n+2)

Answer 1

сначала заметим что

1/2n(2n+2) = 1/4 * 1/n(n+1) = 1/4 *(1/n - 1/(n+1))

1/n(n+1) = (n + 1 - n) = (n+1)/n(n+1) - n/n(n+1) = 1/n - 1/(n+1)  

1/2x4+1/4x6+...1/2n(2n+2) = 1/4*( 1/1*2 + 1/2*3 + .....+ 1/n(n+1)) = 1/4*(1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ..... + 1/n - 1/(n+1)) = 1/4 *( 1 - 1/(n+1)) = 1/4 * (n+1-1)/(n+1) = 1/4*n/(n+1) = n/(4(n+1))