Question

3/1*2+4/2*3+5/3*4+...+n+2/n(n+2)-(1/2+1/3+...1/n)<2,oricare n€ N stelat





Va rog,ajutati-ma!!!

Answer 1

n≠ 0 ca sa aiba sens fractiile
descompunem fiecare termen intr-o suma de 2 termeni
(n+2)/n(n+1)= (n+1)/n(n+1)+ 1/n(n+1)= 1/n+ 1/n(n+1)
deci vom avea
1/1*2+ 1/1+ 1/2*3+ 1/2+1/3*4+ 1/3+...+ 1/n(n+1)+ 1/n- (1/2+1/3+..+1/n)=
reducem termenii asemenea
= 1/2+ 1+1/2*3+ 1/3*4+ ...+ 1/n(n+1)= 
descompunem termenii in diferenta de 2 termeni dupa modelul
1/n(n+1)= 1/n- 1/(n+1) si obtinem
=1/2+ 1+ 1/2- 1/3+ 1/3- 1/4+...+1/n- 1/(n+1)=
=2- 1/(n+1) < 2 pentru orice n natural diferit de zero

Frumos exercitiu!

O zi buna!