"
"
"
1
" Ä… Ä… " R
nÄ…
n=1
"
"
"
"
"
"
"
1
.
n2 + 5n + 6
n=1
" " "
1 1 (n + 3) - (n + 2)
= = =
n2 + 5n + 6 (n + 2)(n + 3) (n + 2)(n + 3)
n=1 n=1 n=1
" "
n + 3 n + 2 1 1
= - = - .
(n + 2)(n + 3) (n + 2)(n + 3) n + 2 n + 3
n=1 n=1
n Sn n
n n
1 1 1 1
an = - Sn = a1 + a2 + . . . + an = ak = -
n+2 n+3 k+2 k+3
k=1 k=1
k = 1, . . . n
1 1 1 1 1 1 1 1 1 1 1 1
Sn = - + - + - + - + - . . . - + - +
3 4 4 5 5 6 6 7 7 n n n + 1
k=1 k=2 k=3 k=4 k={5,6,...,n-3} k= n-2
1 1 1 1
+ - + - .
n + 1 n + 2 n + 2 n + 3
k= n-1 k= n
1 1
Sn = - Sn
3 n+3
"
1 1 1 1 1
S = lim Sn = lim - = < " =
3 n+3 3 n2+5n+6 3
n" n"
n=1
"
1 1
-
n + 2 n + 1
n=1
"
4n
n=1
"
1 + 2 + 3 + . . . + n
.
n2 + 1
n=1
1+n
n
1 + 2 + 3 + . . . + n n2 + n
2
an = = = .
n2 + 1 n2 + 1 2n2 + 2
n2 + n 1
lim an = lim = = 0.

n" n"
2n2 + 2 2
"
n
n2 + 1
n=1
n
lim an = lim = 0.
n" n"
n2 + 1
"
3n2+2
n2 + 2
n2
n=1
"
3n
4n + 6n
n=1
"
n!(n2 + 1)
.
nn
n=1
n!(n2+1)
an =
nn
n
(n+1)! (n+1)2+1
( )
an+1 (n + 1)! ((n + 1)2 + 1) nn
(n+1)n+1
lim = lim = lim =
n!(n2+1)
n" n" n"
an (n + 1)n+1 n!(n2 + 1)
nn
(n + 1)! (n + 1)2 + 1 nn
= lim =
n"
n! n2 + 1 (n + 1)n+1
(n + 1) n! n2 + 2n + 2 nn
= lim .
n"
n! n2 + 1 (n + 1)n(n + 1)
n! (n + 1)
n n -1
an+1 n2 + 2n + 2 n n2 + 2n + 2 n + 1 1
lim = lim · = lim · = e-1 = < 1.
n" n" n"
an n2 + 1 n + 1 n2 + 1 n e
“!
“!
1
e
"
n!(n2+1)
nn
n=1
"
(n + 1)!
2n
n=1
"
(2n)!
n2n
n=1
"
3
lnn 2 + .
n
n=1
3
an = lnn 2 +
n
n
3
2 + > 1
n
" 3 3
n
n
lim an = lim lnn 2 + = lim ln 2 + = ln 2 < 1.
n" n" n"
n n
"
3
lnn 2 +
n
n=1
"
n
2n + 3n
n=1
"
n2
n + 2
n + 1
n=1
Ä„ 2
" x " 0, x d" sin x d" x
2 Ä„
" x " (0, 1) x d" tg x d" 2x
" x e" 1 a > 1 loga x d" x - 1 < x
1
" x " (0, 1) cos x e"
2
"
nn+1 1
2
· tg .
n+1
n
2
(n2 + 3n + 2009)
n=1
n
1 1 1 4
2
tg .
n2 + 3n + 2009 n2 n n2
" " " "
nn+1 1 nn+1 4 nn+1 4 1
2
· tg · = · = 4 · .
n+1 n+1
n n2 nn+1 n2 n2
2 2
(n2 + 3n + 2009) (n2)
n=1 n=1 n=1 n=1
"
1
4 ·
n2
n=1
"
nn+1 2 1
· tg
n+1
n
2
(n2+3n+2009)
n=1
"
"
n
.
n + 3
n=1
"
n
an =
n+3
n
n 3
" "
n n
= an.
n + n n + 3
" "
n n
1
"
= = n 3
n+n 2n 2 n
"
" "
1 n
" .
2 n n + 3
n=1 n=1
" " "
1 1 1 1 1
" = " = ,
1
2 n 2 n 2
2
n
n=1 n=1 n=1
1
2
" "
n
n+3
n=1
"
ln n
.
4n
n=1
ln n
an =
4n
n n " N
ln n n
.
4n 4n
"
n
4n
n=1
n 1
n
lim = < 1.
n"
4n 4
"
ln n
4n
n=1
"
1 1 1
"
n2 · cos · sin · tg .
n n3 n
n=1
1 1 1
"
" (0, 1], " (0, 1], " (0, 1].
n n3 n
1 1 1
"
an = n2 cos sin tg
n n3 n
n
1 1 1 1 2
" "
cos 1, sin tg .
n n3 n3 n n
" " " "
1 1 1 1 2 1 1
" "
n2 · cos · sin · tg n2 · · = 2 · " = 2 · .
3
n n3 n n3 n n n
2
n
n=1 n=1 n=1 n=1
"
1 3
2 ·
3
2
2
n
n=1
"
1 1 1
"
n2 cos sin tg
n n3 n
n=1
"
1 1 1
"
n2 · cos · sin · tg ,
n n2 n
n=1
1 1 1
an = n2 cos sin tg "
n n2 n
n
1 1 1 1 2
" "
cos 1, sin tg .
n n2 n2 n n
" " " "
1 1 1 1 2 1 1
" "
n2 · cos · sin · tg n2 · · = 2 · " = 2 · .
1
n n2 n n2 n n
2
n
n=1 n=1 n=1 n=1
"
1
2 ·
1
2
n
n=1
1
2
n
1 1 1 1 1 1
" "
cos , sin tg .
n 2 n2 2n2 n n
" " " "
1 1 1 1 1 1 1 1 1 1
" "
n2 · cos · sin · tg n2 · · · = · " = · .
1
n n2 n 2 2n2 n 2 n 2
2
n
n=1 n=1 n=1 n=1
"
1 1 1
·
1
2 2
2
n
n=1
"
1 1 1
"
n2 cos sin tg
n n2 n
n=1
"
n2 1
· sin
3n n
n=1
"
"
1
"
n · tg
3
n2
n=1
"
1 1
· cos
n n
n=1
"
n
.
n2 + 1
n=1
n
an =
n2+1
n
1
bn = {an} {bn}
n
n
an n2
n2+1
lim = lim = lim = 1 " (0, +").
1
n" n" n"
bn n2 + 1
n
" "
an bn
n=1 n=1
" "
an <" bn
n=1 n=1
" "
1
bn =
n
n=1 n=1
" "
n
an =
n2 + 1
n=1 n=1
"
2 3
"
(2n2 + 3n + 2011) · sin · tg .
n n3
n=1
2 3 2
3
" "
(2n2 + 3n + 2011) · sin · tg sin
tg
2n2 + 3n + 2011
n n3 n
n3
lim = lim · · = 2 " R+.
2 3 2 3
n" " n" "
n2 · · n2
n n3 n n3
“!
“!
“!
2
1
1
" "
2 3 2 3
" "
(2n2 + 3n + 2011) · sin · tg <" n2 · · .
n n3 n n3
n=1 n=1
" " "
2 3 1 1
" "
n2 · · = 6 · = 6 ·
3
n n3 n n
2
n
n=1 n=1 n=1
3
2
"
2 3
"
(2n2 + 3n + 2011) · sin · tg
n n3
n=1
"
1 1 2
n · cos · sin
4 n n3
n=1
"
" "
1
· n + 3 - n - 3
4n
n=1
" "
"
tg n · sin 2 n
(-1)n .
3n
n=1
"
" " " " "
sin n
tg n · sin 2 n = " · 2 sin n · cos n = 2 sin2 n,
cos n
"
sin2 n 1.
" " " "
" "
tg n · sin 2 n " 2 sin2 n " sin2 n 1
(-1)n = (-1)n = 2 · 2 · .
3n n=1 3n n=1 3n 3n
n=1 n=1
"
1 1
q =
3n 3
n=1
"
1 1
1 1
3 3
= = = < "
3n 1 2 2
1-
3 3
n=1
" " "
(-1)n tg n·sin 2 n
3n
n=1
" " "
(-1)n tg n·sin 2 n
3n
n=1
"
1
(-1)n .
log(2n + 1)
n=1
n log(2n + 1) 2n
" " " "
1 1 1 1 1
(-1)n = = .
log(2n + 1) n=1 log(2n + 1) 2n 2 n
n=1 n=1 n=1
"
1
n
n=1
"
(-1)n 1
log(2n+1)
n=1
"
(-1)n 1
log(2n+1)
n=1
1
an =
log(2n+1)
1
lim an = lim = 0
log(2n+1)
n" n"
an
1 1 log(2n + 1) - log(2n + 3)
an+1 - an = - = .
log(2n + 3) log(2n + 1) log(2n + 1) · log(2n + 3)
an+1 - an < 0.
"
(-1)n 1
log(2n+1)
n=1
"
1
(-1)nn sin .
n
n=1
1
an = (-1)nn sin
n
1
sin
1
n
lim n · sin = lim = 1,
1
n" n"
n
n
"
1
lim an = 0 (-1)nn sin

n
n"
n=1
"
n + 1
(-1)n
n2 + 1
n=1
"
1
"
(-1)ntg
n n
n=1
n
(-1)n
Sn = .
n
Sn = a1 + a2 + a3 + . . . + an-2 + an+1 + an Sn
Sn-1
(-1)n (-1)n-1 1 -1 1 1
an = Sn - Sn-1 = - = (-1)n - = (-1)n + =
n n - 1 n n - 1 n n - 1
n - 1 + n 2n - 1
= (-1)n = (-1)n .
n(n - 1) n(n - 1)
(-1)n
lim Sn = lim = 0.
n" n"
n
"
2n - 1
(-1)n = 0.
n(n - 1)
n=1
n Sn
n + 2
Sn =
nn+ 1
2 - 1
Sn =
2n
0, 12(123)
12 123 123 3 123 1
0, 12(123) = + + + . . . = + 1 + + . . . =
100 100000 100000000 25 100000 1000
"
n
3 123 1 3 123 1 3 123 1
= + = + · = + · =
1 999
25 100000 1000 25 100000 - 25 100000
1
1000 1000
n=0
3 123 1000 3 123 1 3 41 1 4037
= + · = + · = + · = .
25 100000 999 25 100 999 25 100 333 33300
0, 2444 . . .
0, 2(25)