Question

Demonstrati ca: 1/2 la puterea a 2 + 1/3 la puterea a 2 +........1/2011 la puterea a 2< 2010/2011

Answer 1

2010/2011=1-1/2011= 1-1/2+1/2-1/3+1/3-1/4+....+1/2009-1/2010+1/2010-1/2011


(1/2)² =1/4<1-1/2=1/2
(1/3)²= (1/3)* (1/3)<(1/2)*(1/3)=1/(2*3)=1/2-1/3
(1/4)²= (1/4) *(1/4)<(1/3)*(1/4)= 1/(3*4)=1/3-1/4
......................................................................
(1/2011)²= (1/2011) * (1/2011)< (1/2010)* (1/2011)=1/2010-1/2011
 
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(1/2)²+ (1/3)²+...+ (1/2012)²<1-1/2+1/2-1/3+.....++1/2009-1/2010+1/2010-1/2011
(1/2)²+ (1/3)²+...+ (1/2012)²<1-1 /2011=2010/2011

Obs
(1/k)²=1/k², am folosit o scriere mai avantajoasa in acest caz