Question

Vă rog mult, este urgent. Dau 50 de puncte. xx

Answer 1

separam in 2 semisume folosidu-ne de asociativitatea adunarii

incepem cu cei cu numitorii impari

1/1*3=1/2(1/1-1/3) *se poate verifica
1/3*5=1/2(1/3-1/5)
...............................
1/99*101=1/2(1/99-1/101)

adunamd termen cu termen obtinem semisuma
1/2(1/1-1/101)=(1/2)*(100/101)

luam acum pecei cu numitorii pari
1/2*4=1/2(1/2-1/4) poti verifica **
1/4*6=1/2(1/4-1/6)
................................

1/98*100=1/2(1/98-1/100)
adunand termen cu termen, obtinem semisuma
1/2(1/2-1/100)=(1/2)*((50-1)/100)=(1/2)*(49/100)

atunci intrega suma este
(1/2) (100/101+49/100)=(1/2)*((100*100+49*101)/10100)=

=(1/2)((10000+4949)/10100)=14949/20200 =0,740049505 un numar destul de apropiat de 3/4


Obs
 *
(1/2) *(1/1-1/3) =(1/2)* (3-1)/3=1/3
(1/2) *(1/3-1/5)=(1/2)*(5-3)/15=1/15
(1/2)*(1/99-1/101)=(1/2)*(101-99)/99*101 =1/99*101

**
(1/2) (1/2-1/4)=(1/2)* (4-2)/2*4=1/2*4
(1/2) (1/4-1/6)=(1/2)(6-4)/4*6=1/4*6
.................................................................

(1/2)(1/98-1/100)=(1/2)(100-98)/98*100=1/98*100