Question

Fie E(x)=1/x(x+1)
a)Calculati S=E(1)+E(2)+E(3)+.........+E(2019)
b)Aratati ca S aparține intervalului (0,1)

Answer 1

Explicație pas cu pas:

E(x)=1/x(x+1)=(x+1-x)/x(x+1)=(x+1)/x(x+1)-x/x(x+1)

=1/x-1/(x+1)

Deci am ajuns la E(x)=1/x -1/(x+1)

S=E(1)+E(2)+E(3)+.....+E(2019)=>

S=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+....+(1/2019-1/2020)

Scapam de paranteze:

S=1/1-1/2+1/2-1/3+1/3-1/4+....+1/2019-1/2020

Suma noastra este defapt o suma "telescopica" pt ca observam ca seanuleaza termenii din mijloc dar raman capetele

Deci S=1/1-1/2020=>S=2020/2020-1/2020=>S=2019/2020

S€(0,1)=>0<S<1=>0<2019/2020<1=>

0/2020<2019/2020<2020/2020 (A)

Deci S€(0,1)