Question

1/2^2+2/3^2+3/4^2+......+63/64^2 este mai mic decat 6 ..demonstrati

Answer 1

Fie S1 suma data

1/2*2<1/1*2=1-1/2=1/2

2/3*3<2/2*3=1/3

3/4²<3/3*4=1/4

...........................

63/64²<63/63*64=1/64

deci suma noastra este mai mica decat S2= 1/2+1/3+1/4+...+1/64

dar 1/2+1/3<1/2+1/2=1

1/4+1/5+1/6+1/7<1/4=1/4+1/4+1/4+1/4=1

1/8+1/9+....+1/15<1/8+1/8+..+1/8 de 8 ori=1

1/16+1/17+....+1/31<1/16=1/16+...+1/16 de 16 ori=1

1/32+1/33+...+1/63<1/32=1/32+...+1/32 de 32 de ori=1

deci S1 <S2<1+1+1+1+1=5

probabil a tastat cineva (tu sau cine a dat exercitiul sau cine a redactat culegrea) gresit 6...