Question

sa se demonstreze egalitatea : 1 supra 1x3 + 1 supra 3x5+...+1 supra (2n-1)(2n+1)= n pe 2n+1

Answer 1

1/1x3 + 1 /3x5+...+1/(2n-1)(2n+1)= n pe 2n+1

1/1x3=(1/1-1/3)/2 =1/2 (1/1-1/3)
1/3x5=(1/3-1/5)/2=1/2 (1/3-1/5)
1/(2n-1)(2n+1)=[1/(2n-1)-1/(2n+1)]/2=1/2 [1/(2n-1)-1/(2n+1)]
Sn=1/2[1/1-1/3+1/3-1/5+....+1/(2n-1)-1/(2n+1)]=
=1/2(1/1-1/(2n+1)=
=1/2(2n+1-1)/(2n+1)=
=1/2 × 2n/(2n+1)=
=n/(2n+1)