Question

Aratati ca 1/1^2+1/2^2+1/3^2+....+1/2014^2<2

Answer 1

scazand 1 din suma si din 2, revine a arata ca

1/2² +1/3²+1/4²+...+1/2014²<1
dar 1/2²<1/1*2=1-1/2
1/3²<1/2*3=1/2-1/3
1/4²<1/3*4=1/3-1/4
....................................
1/2013²<1/2012*2013=1/2012-1/2013
1/2014²<1/2013*2014=1/2013-1/2014

insumand
 S<1-1/2014=2013/2014<1  adevarat