Question

A.) Calculeaza suma : S = 1/2 + 1/2*3 + 1/3*4 + ...+ 1/89*90
B.) Arata ca S > 1/2

Answer 1

A)S=1/1·2+1/2·3+...+1/89·90 datorita formulei 1/k(k+1)=1/k-1/(k+1) obtinem
S=1/1-1/2+1/2-1/3+...+1/89-1/90
S=1-1/90
S=90/90-1/90
S=89/90
B) 89/90 SI 1/2(amplificam pe 1/2 cu 45)
89/90 si 45/90 ⇒89/90mai mare decat 45/90 deci S mai mare decat 1/2