Wydział WILiŚ, Budownictwo, sem.3
dr Jolanta Dymkowska
Szeregi liczbowe
Zad.1 Zbadaj z definicji zbieżność szeregów:
" " " "
n

3 1 6 1 1
1.1 1.2 + 1.3 1.4
5n 4n 7 n2+3n+2 (3n-2)(3n+1)
n=1 n=1 n=1 n=1
" "

n(n+2)
n+2
1.5 ln 1.6 ln
n+1 (n+1)2
n=1 n=1
Zad.2 Zbadaj zbieżność szeregów o wyrazach nieujemnych:
n n
" " " " "
n
n
n
3n-1 1 2n n
2.1 2.2 sin 2.3 2.4
4n+5 n 7 6n+1
n=1 n=1 n=1 n=1
" " " 2 "
2
n
-1
4n4 2 n+1 3n
2.5 2.6 e-n! 2.7 2.8 "
n
2
2n n
3+ 2n2 n
( )
n
n=1 n=1 n=1 n=1
2n n n
" " " " "
n

n
ln n 1 1
" "
2.9 2.10 2.11 arcsin 2.12
n en-1 n arctg n!
n=1 n=1 n=1 n=1
" " " "

(n+5) 5n 1·3·...·(2n-1)
3n n! n2n
2.13 2.14 2.15 2.16
7n 3n+1 nn 6n (n!)2 3n n!
n=1 n=1 n=1 n=1
" " " "

(5n n! (n+2)! (2n)! 2n
2n
2.17 2.18 2.19 2.20
(2n)n+1 8n (n!)2 n2n (1+2)(1+22)...(1+2n)
n=1 n=1 n=1 n=1
" " " "


cos2 nĄ
1 Ä„ 4
3
2.21 cos 2.22 1 - cos 2.23 2.24 sin
n n 2n 3n
n=1 n=1 n=1 n=1
" " " " "
" 3
n
Ä„ 1
3 3 2Ä„
" "
2.25 n sin2 1 2.26 n tg 2.27 tg 2.28 tg
n 2n+1 1+ n n n
n=1 n=1 n=1 n=1
"
" " " "

n3 ( 2+(-1)n)n
n ln n ln n
2.29 ( 2 + (-1)n ) 2.30 2.31 2.32
4n 3n n 2n
n=1 n=1 n=1 n=1
"
"
" " " " "

n+1
n+1- n
7n-1 2n+2 3
"
2.33 2.34 2.35 2.36
n
3n2+8 n2 (n+3) n
3
n=1 n=1 n=1 n=1
" " " " "
3
4n3-1 n+1 n+1 1
" " " "
2.37 2.38 2.39 2.40
3
(3n2+8)(5n2-1) n (n+2) n 2n-1 3n+8
n=1 n=1 n=1 n=1

" " " " " "
3
n+2 2n-3 n-12
1 n+1
4
" " "
2.41 2.42 2.43 2.44
4
n n+n n n3+3 n2+6
4n3+n2-1
n=1 n=1 n=1 n=1
"
" " " "

1 1 1 e- n
"
2.45 2.46 2.47 2.48
n ln n n ln2 n n ln n ln(ln n) n
n=2 n=2 n=3 n=1
" " " "

1+ln n 1 n arctg n
"
2.49 2.50 2.51 2.52
4
n en n2+1
n ln n
n=2 n=2 n=3 n=1
Zad.3 Zbadaj zbieżność szeregów:
" " " "

(-1)n (-1)n (-1)n+1 (n+2)
cos nĄ
"
3.1 3.2 3.3 3.4
n4+4 n2+n n3+2n+4
1+ n+1
n=1 n=1 n=1 n=1
" " " "

(-1)n ln n (-1)n+1 (-1)n
1
"
3.5 3.6 3.7 3.8 (-1)n n sin
n 2 n-1 n4+4 n3
n=1 n=1 n=1 n=1
n
" " " "
1

(-7)n (n!)2 sin
(-1)n 10n n!
n
3.9 (-1)n 6n-1 3.10 (-1)n n! 3.11 3.12
9n+4 nn (2n)! n2n-1
n=1 n=1 n=1 n=1
" " " "
nĄ nĄ

sin (2n)! sin
n cos n! n3n cos n
6 2
3.13 3.14 3.15 3.16
n4 n3+3 n2n (3n)!
n=1 n=1 n=1 n=1
1


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