Введение в математический анализ в вопросах и задачах. Анчиков А.М - 38 стр.

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a) x
n
= sin n; b) x
n
= (−1)
n
; c) x
n
= (−1)
n
1
n
; d) x
n
=
=
(−1)
n
n
+
1+(−1)
n
2
.
”ε −n
ε
” n
ε
a) lim
n→∞
3n
n+5
= 3; b) lim
n→∞
cos 3n
n
= 0; c) lim
n→∞
2
n
−7·6
n
3
n
+6
n
= −7;
d) lim
n→∞
3n
2
+2
4n
2
−1
=
3
4
; e) lim
n→∞
5·3
n
3
n
−2
= 5; f) lim
n→+∞
log
n
2 = 0.
”ε − n
ε
”
{x
n
} a :
a) x
n
=
(−1)
n+1
n
, a = 0; b) x
n
=
2+(−1)
n
n
, a = 0; c) x
n
=
n
2
+n−2
3n
2
+2n−4
,
a =
1
3
; d) x
n
=
n
√
c (c > 1), a = 1.
h
a) n
ε
= [
1
ε
]; b) n
ε
= [
3
ε
]; c) n
ε
= [
1
ε
]; d) n
ε
= [
1
log
c
(1+ε)
]
i
.
a) lim
n→∞
n
√
a = 1, (a > 0); b) lim
n→∞
n
√
n = 1;
c) lim
n→∞
(
1
2
·
3
4
· ... ·
2n−1
2n
) = 0; d) lim
n→∞
q
n
= 0, (|q| < 1).
{x
n
}
lim
n→∞
x
n
lim
n→∞
x
n
, a) x
n
= 1 −
1
n
; b) x
n
= (−1)
n
(2 +
3
n
); c)x
n
= (−1)
n
; d) x
n
= n
(−1)
n
; e) x
n
= 1 +
n
n+1
cos
nπ
2
.
[ a) 0, 1; b) − 2, +2; c) − 1, +1; d) lim
n→∞
x
n
=
0, lim
n→∞
x
n
= ∞; e) −1, 0; +1, e) lim
n→∞
x
n
= −1, lim
n→∞
x
n
= ∞ ].
{x
n
}, x
n
=
1
√
n
2
+1
+
1
√
n
2
+2
+ ... +
1
√
n
2
+n
.
z
n
< x
n
< y
n
, z
n
=
n
√
n
2
+n
, y
n
=
n
√
n
2
+1
a) xn = sin n; b) xn = (−1)n ; c) xn = (−1)n n1 ; d) xn =
      n          n
= (−1)
    n
        + 1+(−1)
             2
                   . Êàêèå èç íèõ îãðàíè÷åíû, ñõîäÿòñÿ. Îòâåòû
îáîñíóéòå.
   2. Ïîëüçóÿñü ÿçûêîì ”ε − nε ” äîêàæèòå, ÷òî: (óêàæèòå nε )
         3n                                                        n       n
a) n→∞
    lim n+5          lim cosn3n = 0; c) n→∞
            = 3; b) n→∞                  lim 23n−7·6
                                                 +6n
                                                     = −7;
         3n2 +2                         5·3n
d) lim     2       = 34 ; e) lim         n     = 5; f )         lim logn 2 = 0.
     n→∞ 4n −1                      n→∞ 3 −2                n→+∞

   3. Ñ ïîìîùüþ ”ε − nε ” ðàññóæäåíèé äîêàçàòü ñõîäèìîñòü
ïîñëåäîâàòåëüíîñòè {xn } ê a :
           (−1)n+1                                   n
                                                                                 n2 +n−2
a) xn =       n
                  a = 0; b) xn = 2+(−1)
                   ,                 n
                                        , a = 0; c) xn =                        3n2 +2n−4
                                                                                          ,
                 √
a = 13 ; d) xn = n c (c > 1), a = 1.
h                                                                                  i
    a) nε = [ 1ε ]; b) nε = [ 3ε ]; c) nε = [ 1ε ]; d) nε = [ log (1+ε)
                                                                   1
                                                                        ] .
                                                                 c
                                        √                                √
     4. Äîêàæèòå, ÷òî: a) lim n a = 1, (a > 0); b) lim n n = 1;
                                      n→∞                                  n→∞

c) lim ( 12 · 34 · ... ·   2n−1
                            2n
                                )   = 0; d) lim q n = 0, (|q| < 1).
     n→∞                                       n→∞

   5. Äëÿ ïîñëåäîâàòåëüíîñòè {xn } íàéòè âñå ïðåäåëüíûå òî÷-
êè, lim xn è lim xn , åñëè a) xn = 1 − n1 ; b) xn = (−1)n (2 +
      n→∞              n→∞
3                                                n n
n
  ); c)x n =   (−1) ; d) xn = n(−1) ; e) xn = 1 + n+1
                       n
                                                      cos nπ
                                                           2
                                                             .
[ a) 0,       1; b) − 2, +2; c) − 1, +1; d) lim xn =
                                                                                n→∞
0, lim xn = ∞; e) − 1, 0; +1, e) lim xn = −1, lim xn = ∞ ].
     n→∞                                         n→∞                      n→∞

    6. Íàéòè ïðåäåë ïîñëåäîâàòåëüíîñòè                           {xn },        åñëè     xn =
√ 1    + √n12 +2 + ... + √n12 +n .
 n2 +1


     [ Óêàçàíèå: zn < xn < yn , zn =                 √ n    ,    yn =     √ n      ].
                                                      n2 +n                n2 +1




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