05.1 - Szeregi liczbowe.pdf

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WydziałWILi,Budownictwo,sem.3
drJolantaDymkowska
Szeregiliczbowe
Zad.1Zbadajzdefinicjizbie»no±¢szeregów:
1.1
1 P
5 n 1.2
1 P
1 4 n + 6 7 n 1.3
1 P
n 2 +3n+2 1.4
1
1 P
1
(3n−2)(3n+1)
n=1
n=1
n=1
n=1
1 P
1 P
1.5
ln n+2
n+1 1.6
ln n(n+2)
(n+1) 2
n=1
n=1
Zad.2Zbadajzbie»no±¢szeregówowyrazachnieujemnych:
2.1
1 P
3n−1
4n+5
n
2.2
1 P
sin 1 n n
2.3
1 P
n p n 7
n
2.4
1 P
2 n n
6 n+1
n=1
n=1
n=1
n=1
1 P
1 P
1 P
2 n n+1
n 2
1 P
2.5
( 3+ 2 n ) n 2.6
4n 4
e −n!
2.7
2
2.8
3 n 2 −1
2 n 2 p n
n
n=1
n=1
n=1
n=1
p n
p en−1
2n
n
1
arctgn!
n
1 P
lnn n
n
1 P
1 P
1 P
arcsin 1 p n
2.9
2.10
2.11
2.12
n=1
n=1
n=1
n=1
2.13
1 P
(n+5)5 n
7 n 3 n+1 2.14
1 P
3 n n!
n n 2.15
1 P
6 n (n!) 2 2.16
n 2n
1 P
1·3·...·(2n−1)
3 n n!
n=1
n=1
n=1
n=1
1 P
1 P
1 P
1 P
2.17
(5 n n!
(2n) n+1 2.18
(n+2)!
8 n (n!) 2 2.19
(2n)!2 n
n 2n 2.20
2 n
(1+2)(1+2 2 )...(1+2 n )
n=1
n=1
n=1
n=1
2.21
1 P
cos 1 n
2.22
1 P
1−cos n
2.23
1 P
cos 2 n 3
2 n 2.24
1 P
sin 4 3 n
n=1
n=1
n=1
n=1
1 P
3 p nsin 2 1 n
1 P
1 P
3 p n
1 P
2.25
2.26
ntg
2 n+1 2.27
1+ p n tg 1 p n 2.28
tg 3 2 n
n=1
n=1
n=1
n=1
1 P
1 P
n 3 ( p 2+(−1) n ) n
3 n 2.31
1 P
1 P
2.29
4 n (2+(−1) n ) 2.30
n
lnn
n
2.32
lnn
2 n
n=1
n=1
n=1
n=1
2.33
1 P
7n−1
3n 2 +8
2.34
1 P
2n+2
n 2 (n+3)
2.35
1 P
p n+1− p n
n
2.36
1 P
3
p n+1
3 p n
n=1
n=1
n=1
n=1
1 P
1 P
1 P
3 p n+1
p n 3 p 2n−1 2.40
1 P
2.37
(3n 2 +8)(5n 2 −1) 2.38
4n 3 −1
n+1
p n(n+2)
2.39
1
p 3n+8
n=1
n=1
n=1
4 q n+1
n 3 +3
n=1
1 P
1 P
3 p n+2
n p n+n 4 p n
1 P
1 P
2n−3 p n−12
n 2 +6
2.41
1
p 4n 3 +n 2 −1
2.42
2.43
2.44
n=1
n=1
n=1
n=1
2.45
1 P
1
nlnn
2.46
1 P
1
nln 2 n
2.47
1 P
nlnnln(lnn) 2.48
1
1 P
e − p n
p n
n=2
n=2
n=3
n=1
1 P
1 P
1 P
1 P
2.49
1+lnn
n
2.50
1
n 4 p
2.51
e n 2.52
arctgn
n 2 +1
lnn
n=2
n=2
n=3
n=1
Zad.3Zbadajzbie»no±¢szeregów:
3.1
1 P
(−1) n
n 4 +4
3.2
1 P
(−1) n
1+ p n+1 3.3
1 P
cosn
n 2 +n
3.4
1 P
(−1) n+1 (n+2)
n 3 +2n+4
n=1
n=1
n=1
n=1
1 P
1 P
1 P
1 P
3.5
(−1) n lnn
n
3.6
(−1) n+1
2 p n−1 3.7
(−1) n
n 4 +4
3.8
(−1) n nsin 1 n 3
n=1
n=1
n=1
n=1
3.9
1 P
(−1) n 6n−1
9n+4
n
3.10
1 P
(−1) nn!
n n 3.11
1 P
(−1) n 10 n n!
(2n)! 3.12
1 P
(−7) n (n!) 2 sin 1 n
n 2n−1
n=1
n=1
n=1
n=1
3.13
1 P
sin n 6
n 4 3.14
1 P
ncosn!
n 3 +3 3.15
1 P
(2n)!sin n 2
n 2n 3.16
1 P
n 3n cosn
(3n)!
n=1
n=1
n=1
n=1
1
3
n
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