" "
"
"
3+ n
1 n2+4
"
a) an = , b) bn = n + 2 - n, c) cn = , d) dn = ,
1-3n 3n-2
4n2+3n-2n

" " "
"
2
3n-2 1+2n2- 1+4n2
"n +5-n
e) en = , f) fn = , g) gn = n2 - 1 - n, h) hn = ,
n+10 n
n2+5+n
"
" " "
(3- n)2
1
"
i) in = n2 + n - n, j) jn = , k) kn = 2n - 1 - n + 7, l) ln = ,
5+4n
n2+6-n
" " "
"
2+ n
3n2+5-n 4n2+9
m) mn = 9n2 + 1 - 3n, n) nn = , o) on = , p) pn = .
3n+1 2n+1 1-2n
n
4n-1+5 3 2n+1-1 5·32n-1 4·3n+2+2·4n
a) an = , b) bn = · , c) cn = , d) dn = ,
22n-7 2 3n+1-1 4·9n+7 5·2n+4n+2
2n+(-2)n
4-6n 1-7·32n 3n+5n
e) en = , f) fn = , g) gn = , h) hn = .
8n-16n 22n 4·9n+7 6n-2n
-2n n2 3n+2 n
1 n2+6 n+5 2
a) an = 1 + , b) bn = , c) cn = , d) dn = 1 + ,
n n2 n
n n 2n+1 n n
1 1 n+2 1
e) en = 1 - , f) fn = 1 + , g) gn = , h) hn = 1 - ,
n n2 n n2
-n+4 n2 3n-4 2n2
+5
4 n2+2 2n+5 n2+3
i) in = 1 - , j) jn = , k) kn = , l) ln = ,
n 2n2+1 3n+7 n2+1
2n n 2n2 n
-4
n n-1 4 3n+2
m) mn = , n) nn = , o) on = 1 + , p) pn = .
n-6 n+2 n2+3 3n+1
n
"
n
n n 1
a) an = Ä„n + 3n + (3, 14)n, b) bn = 5 · 77 + 7 · 5n, c) cn = + n,
3
" " n
n n n 1
d) dn = 2 · 3n + 4 · 7n, e) en = 3n + sin n, f) fn = + 2n,
2
" " "
n n n
g) gn = en + 3n, h) hn = + n2, i) in = n2 + 5n - 10,
2n
" 1 n 2 n 3 n
n n n
j) jn = 10n + 9n + 8n, k) kn = + + , l) ln = 2n + (-1)n.
2 3 5
"
"
10000n 5n2+n-3
1) ; 2) ; 3) + 2 - n;
n2+1 15n2-7
n
"
"
" "
n2+1+ n
n+1
3
"
4) 3n2 + 2n - 5 - n 3; 5) ; 6) ;
3
3n+2
n- n2+8
" "
" "
3
7) n3 + 5n - n; 8) n( n + 2 - n); 9) (3n+2)4;
10+n
"
3
(-1)n n sin(n!)
sin n
10) ; 11) ; 12) ;
2n-1 n n
13) (-1)n+1(2)n; 14) (-3)n; 15) (-5)n;
3 5 3
2n+(-1)n
3n+1+2n+1 3·22n+2-10
16) ; 17) ; 18) ;
3n+2n 2n+1 5·4n+4+3
(-2)n+3n
2n+1-1 2n+2n!
19) (3)n · ; 20) ; 21) ;
2 3n+1-1 (-2)n+1+3n+1 3n+5
22)(n+1)!+(n+2)!.
n!+(n+3)!

" "
n n n sin 2n
n
1) 2n + 3n; 2) 4n + 6n; 3) 3n + (1)n + 9n; 4) ;
2
2
n +1 n
2n n+sin n 1 1 1 1 5
" " " " "
5) +
2
n2+1 n2+2 n2+3 n2+n
n +sin 6) n ; n 7) n + + . . . + ; 8) 1 + n n ;
nn+n;
n+9 1 3 n
9) 12) ;
n n2 n+1
n ; n 10) 1 - ; 11) 1 - ;
6
ln(1+ )
n2+2
n
13) ; 14) ; 15) n(ln(n + 1) - ln n).
1
2n2+1
n
1 3 n nĄ
1) 1 - ; 2) (-1)n-1(2 + ); 3) 1 + cos ;
n n n+1 2
n(n-1)
1 1 3 1 7 1 2n-1
2
4) 1 + 2(-1)n+1 + 3(-1) ; 5) {1, , , , , , . . . , , , . . .}.
2 2 4 4 8 8 2n 2n
n
limn" n+1 = 1
an = {0, 1, 0, 1, . . .}
an = {(-2)n}
"
limn"( n2 + 1 - n) = 0
(-1)n+1
an = {(-1)n + }
n
limn"(n2n ) = 0
3
+1
limn" (-1)n = 0
n
limn" xn = a limn" |xn| = |a|
an+1+bn
an+bn
{an} {bn} a1 = a b1 = b an+1 = bn+1 =
2 2
limn" an = limn" bn
{an} limn"(an+1 - an) = 0
"
"
"
"
"
n 2 n
"
n
limn" n = 1
1
limn" "n! = 0
n
limn" an = 0
n!


"
1 1 1 1
1) an = (1 + ); 2) (1 + )(1 + ) . . . (1 + ); 3) 2 + 2 + . . . + 2;
n 2 4 2n
sin 1 sin 2 sin n cos 1! cos 2! cos n! 1 1 1
4) + + . . . + ; 5) + + . . . + ; 6) 1 + + + . . . + .
2 22 2n 1·2 2·3 n(n+1) 22 32 n2
1 1
xk = 1 + + . . . +
2 k
1 1 1
xn = 1 + + + . . . +
ln 2 ln 3 ln n
an 0 {bn} anbn 0
" "
a1 = c an+1 = c + an
1
b1 = 0 b2 = 1 bn = (bn-1 + bn-2)
2