حلول تمارين المحاضرة الخامسة

Determine whether the sequence converges or diverges.
1.{(3+ln??n^n ?)/n^2 }
lim?(n??)??(3+ln??n^n ?)/n^2 ?=lim?(n??)??3/n^2 ?+lim?(n??)??(n ln?n)/n^2 ?=0+lim?(n??)??ln?n/n?=0 converges to 0
2. {(( 3 )/n)^(1/n) }
lim?(n??)??(( 3 )/n)^(1/n) ?=lim?(n??) ?(n&3)/?(n&n)=(lim?(n??) ?(n&3))/(lim?(n??) ?(n&n))=1 converges to 1
3. {((n+5)/n)^n }
lim?(n??)??((n+5)/n)^n ?=lim?(n??)??(1+5/n)^n ?=e^5 converges to e^5
4. {?(n&4^n n)}
lim?(n??) ?(n&4^n n)=lim?(n??) ?(n&4^n )×lim?(n??) ?(n&n)=4×1=4 converges to 4
5. {(n^5+2n)/(3n^4+n^2 )}
lim?(n??) (n^5+2n)/(3n^4+n^2 )=lim?(n??) ((n^5+2n)/n^5 )/((3n^4+n^2)/n^5 )=lim?(n??) (1+2/n^4 )/(3/n+1/n^3 )=? diverge
6. {8^(n+1)/n!}
lim?(n??) 8^(n+1)/n!=8 lim?(n??) 8^n/n!=8×0=0 converges to 0
7.{(e^n+n^2)/(e^n-2n^2 )}
lim?(n??) (e^n+n^2)/(e^n-2n^2 )= lim?(n??) (e^n+2n)/(e^n-4n)= lim?(n??) (e^n+2)/(e^n-4)= lim?(n??) e^n/e^n =1 converges to 1




8. {?(n(n+2) ) -n}
lim?(n??) ?(n(n+2) ) -n= lim?(n??) ?(n^2+2n) -n×(?(n^2+2n)+n)/(?(n^2+2n) +n)=lim?(n??) (n^2+2n-n^2)/(?(n^2+2n) +n)=lim?(n??) 2n/(?(n^2+2n) +n)=lim?(n??) 2/(?(1+2/n) +1)=1 converges to 1
9. {(3^n/(2n+1))^(1?n) }
lim?(n??) (3^n/(2n+1))^(1?n)= lim?(n??) ?(n&3^n )/?(n&2n+1)=3/?lim?(n&2n+1)??(n??) =3/1=3 converges to 3

10. {(3n^2-ln?n)/(n^2+3n^(3?2) )}
lim?(n??) (3n^2-ln?n)/(n^2+3n^(3?2) )=lim?(n??) (6n-( 1 )/n)/(2n+9/2 ?n)=lim?(n??) (6+( 1 )/n^2 )/(2+9/(4?n))=(6+0)/(2+0)=3 converges to 3
11. {((n-3)/n)^n }
lim?(n??)??((n-3)/n)^n ?=lim?(n??)??(1+(-3)/n)^n ?=e^(-3) converges to e^(-3)