шамин с сдо
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1.0 |
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0.8 |
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1 −n |
0.6 |
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0.4 |
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0.2 |
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0.0 |
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n |
30 |
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50 |
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an = |
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| sin k|. |
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X |
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k=1 |
2k |
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| sin k| |
≥ 0 |
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| sin k| |
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2. |
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n |
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k=1 |
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≤ k=1 |
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{an}
E
e
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n |
e = n→∞ 1 + n |
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lim |
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(1 + h)n > 1 + nh, h > 0, n ≥ 2,
n N
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n+1 |
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xn = 1 + |
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n |
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xn > 1 + n + 1 > 2. n
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1 + |
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n |
xn |
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n1 |
n |
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xn−1 |
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1 + n−1 |
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{xn}
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n2 |
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n + 1 |
(n + 1)(n − 1) |
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n |
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= n + 1 1 + n2 − 1 > |
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n |
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1 |
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> |
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1 + |
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n3 + n2 − n |
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> 1. |
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n2 − 1 |
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n + 1 |
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n3 + n2 − n − 1 |
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{xn} |
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[2, 4] |
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n→∞ |
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n |
n |
n→∞ |
1 + n1 |
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lim xn |
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n→∞ |
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nlim 1 + n1 |
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e = lim |
1 + |
1 |
= lim |
xn |
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n→∞ |
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= lim xn. |
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→∞ |
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e
e = 2.7182818284590452353602874713527 . . ..
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{an} |
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{bk} |
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{an} |
k |
nk |
nk1 < nk2 |
k1 < k2 |
bk = ank
an = (−1)n.
1
−1
{xn}
[a, b]
xn [a, b].
[a1, b1]
b − a b1 − a1 = 2 .
xn1 [a1, b1]
[a1, b1]
{xn} |
[a2, b2] |
xn2 [a2, b2]
xnk [ak, bk].
b |
k − |
a |
k |
= |
b − a |
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2k |
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y
xnk
ak ≤ xnk ≤ bk
lim ak = lim bk = y.
k→∞ k→∞
lim xnk = y.
k→∞
an = cos n,
−1 ≤ an ≤ 1,
{xn}
ε > 0
N = N(ε)
|xn − xm| < ε,
n, m > N(ε)
{xn} |
a |
ε > 0
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N = N(ε) |
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|xn − a| < |
ε |
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2 |
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n > N(ε) |
n, m > N(ε) |
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|xn − xm| = |xn − a + a − xm| ≤ |xn − a| + |xm − a| < |
ε |
+ |
ε |
= ε. |
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2 |
2 |
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{xn}
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{xnk } |
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a |
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{xn} |
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ε > |
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N = N(ε) |
nk > N(ε) |
n, m > N(ε) |
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|xnk − a| < |
ε |
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2 |
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ε |
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|xn − xm| < |
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{xn}
ε ε
|xn − a| = |xn − xnk + xnk − a| ≤ |xn − xnk | + |xnk − a| < 2 + 2 = ε.
lim xn = a
n→∞
1 an = n .
ε > 0
N
N > 2ε .
n, m > N
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1 |
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≤ |
1 |
1 |
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n |
− m |
≤ n |
+ m |
N |
+ N |
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lim αn = 0.
n→∞
= N2 ≤ ε.
{αn}
{an} |
c |
{αn}
an = c + αn.
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{an} |
c |
αn = an − c. |
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ε > 0 |
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N = N(ε) |
|an − c| = |αn| < ε, n > N(ε), |
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{αn} |
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{an} |
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an = c + αn, |
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αn |
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an |
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c |
ε > 0 |
N = N(ε) |
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|αn| = |an − c| < ε, n > N(ε). |
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lim an = c |
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n→∞ |
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1 |
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an = sin |
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n |
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lim sin |
1 |
= 0, |
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n→∞ |
n |
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{an}
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an = |
2n2 |
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n2 + 1 |
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lim |
= 2 |
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n→∞ |
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an = 2 + αn, |
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αn |
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2n2 |
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= |
2n2 + 2 − 2 |
= 2 |
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2 |
= 2 + α , |
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n2 + 1 |
n2 + 1 |
− n2 + 1 |
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n |
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αn = −n22+1
{βn}
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M > 0 |
N = N(M) |
n > N(M) |
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|βn| > M. |
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nlim βn = ∞ |
n > N(M) |
→∞ |
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βn > M, |
lim = +∞
n→∞
βn < −M,
lim = −∞
n→∞