Odpowiedzi

2009-12-11T18:27:16+01:00
1)
a_n=2-1/n
a_(n+1)= 2- 1/(n+1)

a_(n+1)- a_n=
2- 1/(n+1) -2 +1/n=
1/n - 1/(n+1) =
(n+1-n)/[n(n+1)]=
1/(n²+n)
1>0 ∧ n²+n>0
ciąg rosnący

2)a_n= -3/(n+1)
a_(n+1)= -3/(n+2)
a_(n+1)- a_n= -3/(n+2) + 3/(n+1)=
(-3n-3+3n+6)/[(n+2)(n+1)]=
3/(n²+3n+2)
3>0 ∧ n²+3n+2>0
ciąg rosnący

3) a_n= n/(n²+1)
a_(n+1) = (n+1)/[(n+1)²+1]
a_(n+1) - a_n = (n+1)/(n²+2n+2) - n/(n²+1)=
[(n+1)(n²+1)-n(n²+2n+2)]/[(n²+2n+2)(n²+1)]=
(n³+n²+n+1-n³-2n²-2n)/[(n²+2n+2)(n²+1)]=
(-n²-n)/[(n²+2n+2)(n²+1)]=
-(n²+n)/[(n²+2n+2)(n²+1)]
n²+n > 0, -(n²+n) <0 , (n²+2n+2)(n²+1) > 0
ciąg malejący