Toshkent davlat transport universiteti
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3-masala. Qatorlarning yaqinlashish masalasini Dalamber alomati yordamida yeching. 3.1. 1
3 ...
... 2! 5! 8! (3 1)! n n + + + + + −
3.2. 2 3 3 3 3 3 ...
... 3! 5! (2 1)!
n n + + + + + −
3.3. ( )
1 9 14 19 5 1 4 ...
... 5 5 5 5 5
n − − + + + + + +
3.4. 3 4 3 3 3 1 3 3 3 3 ...
... 3 5 7 (2 1) n n + +
+ + +
+ −
3.5. sin 2sin 3sin
... sin
... 3 9 27 3
n + + + +
+
3.6. 2 1 4 9 16 ... ( 1) ... 4 8 16 2 n tg tg tg n tg + + + + + + +
3.6. 2 1 4 9 16 ... ( 1) ...
4 8 16 2 n tg tg tg n tg + + + + + + +
3.7. 3 3 5
3 5 7 3 5 7 ... (2 1) ...
... 2 2 5 2 5 8 2 5 8 ... (3 1)
+ + + + +
+ −
3.8. 1 5 1 5 9
1 5 9 ...(4 3) 1 ... ...
1 4 1 4 7
1 4 7 ...(3 2)
n − + + + +
+ −
3.9. 1 1 5 ! n n n n − =
3.10. 1 4 ( 1)! n n n n = +
3.11. 2 1 (2 )! ( !)
n n n =
3.12. 1 (2 1)! (4 3) 3
n n n n = + +
3.13.
2 1 (2 )! ( !) n n n =
3.14. 1 1 (2 1)! (3 4) 3 n n n n − = − +
3.15. 1 1000
! n n n =
3.16. 1 !
n n n =
3.17. 2 2 1 (n!) 2
n =
3.18. 1 2 ! n n n n n =
3.19.
2 2 1 (n!) 3
n =
3.20. 1 3 ! n n n n n =
3.21.
1 10 ! n n n =
3.22. 1 3 ! n n n =
3.23.
1 1 3 n n n n + =
3.24. 1 (5 ) (2 1)!
n n n n = +
3.25. 2 3 3 3 3 5 5 5 5 ... ...
4 7 (3 2) n n + + + + + −
3.26. 2 4 8 2 sin 2sin 3sin
... sin
... 3 9 27 3
n n + + + +
+
3.27. 2 1 2 4 8 2 4 9 16 ... (
1) ...
9 27 81 3 n n tg tg tg n tg + + + + + + +
3.28. 2 1 4 9 16 ... ...
4 16 64 4 n tg tg tg n tg − + + + + +
3.29. 2 2 6 2 6 10 2 6 10 ... (4 2) ...
... 1 1 6 1 6 11 1 6 11 ... (5 4)
− + + + +
+ −
3.30. 2 2 7
2 7 12 2 7 12 ...(5 3) ...
... 1 1 4 1 4 7 1 4 7 ...(3 2)
− + + + +
+ −
4-masala. Qatorlarning yaqinlashish masalasini Koshi yoki integral alomati yordamida yeching. 4.1. 2
1 3 5 2 1 ... ... 5 9 13 4 1 n n n − + + + + + +
4.2. 3 2 2 2 3 4 5 1 1 ... ...
5 8 11 3 1
n n + + + + + + +
−
4.3.
2 4 6 2 1 ... .... 5 10 1 n n + +
+ + + + 4.4.
2 3 3 3 3 ... ... 5 8 13 4
+ + + +
+ +
4.5. 2 4 6 2 1 1 1 1 ... ...
ln 2 ln 5
ln 8 ln (3
1) n n + + + + + − 4.6.
1 3 2 2 2 1 1 1 1 ... ...
2 4 6 2 n arctg arctg arctg arctg n + + + + +
4.7. 2 2 2 2 1 1 1 1 ... ...
3ln 3 4ln 4
5ln 5 ( 2)ln ( 2) n n + + + + + + +
4.8. 1 1 1 1 ...
... 2 1 ln 2
2 1 ln 3 4 1 ln 4
1 ln n n + + + + + + + + + 4.9.
2 4 9 2 4 6 2 ...
... 3 5 7 2 1 n n n
+ + + + +
+
4.10.
2 4 9 2 3 3 4 5 2 3 3 3 ... 3 ...
4 5 6 3 n n n n + + + + + +
+
4.11.
1 1 1 1 ...
... 2 7 12 5 3 n + +
+ + + − 4.12.
1 1 1 1 ...
... 5 8 11 3 2 n + + + + + + 4.13.
4 2 4 6 2 ... ... 5 20 85 4
n + + + + + + 4.14. 2 1 1 1 1 ... ...
3 8 15 2 n n + +
+ + + + 4.15.
( 1) 1 1 1
n n n − = − +
4.16. 1 1 ln ( 1) n n n = +
4.17. 2 2 1 1
n n n = + −
4.18.
2 1 5 1 n n n n = +
4.19. 2 1 1 1
n n n n − = − +
4.20. 3 2 1 2 3 n n n n = + + 4.21.
2 1 4 (ln ) n n n =
4.22. 1 1 7 4
n = −
4.23. 2 1 ln(ln ) n n n =
4.24. 2 2 6 (ln )
n n n =
4.25. 2 1 1 ln ( 1)
n n = +
4.26. 1 1 ( 1)ln 2
n n n = +
4.27. 2 1 2 3 1 n n n n = + +
4.28.
1 1 3 n n n e − − = 4.29.
1 arcsin
4 n n n n =
4.30. 2 2 3 ( 1)ln n n n n = +
5-masala. Quyidagi ishoralari o‘zgaruvchi qatorlarni yaqinlashishga tekshiring. Yaqinlashuvchi qatorlar uchun absolyut yoki shartli yaqinlashuvchi ekanligini aniqlang. 5.1.
( ) ( ) = + + − 0 1 2 2 1 2 1 1 n n n n .
5.2. ( ) ( ) = − + + − 1 1 1 ln 3 1 n n n n . 5.3. ( ) ( ) ( ) = + − + − 1 1 1 3 ...
5 2 1 2 ...
5 3 1 n n n n .
5.4.
( ) = − + + − 1 2 1 1 1 2 1 n n n n . 5.5. ( ) ( ) = − + − 1 2 1 3 1 1 1
n n n .
5.6. ( ) ( ) = − + − 1 1 3 1 1
n n n .
5.7. ( ) (
) = + − 1 2 ! 1 1
n n n .
5.8. = 1 3 cos n n n n . 5.9. ( )
( ) ( ) = + + − − 1 1 5 2 ... 9 7 2 3 ... 4 1 1 n n n n .
5.10.
( ) ( ) 1 1 1 2 ln ln 2 n n n n n n + − = . 5. 11.
( ) =1 10 ln sin
n n n .
5.12. ( )
= + + − 1 5 1 1 3 1 n n n n . 5.13. ( ) ( ) 1 1 ln
2 n n n n − + = .
5.14.
( ) ( ) = + − 1 3 ! 1 1
n n n . 5.15. ( ) ( )
= − 1 sin 1 n n n n n n .
5.16. ( ) = + − 1 2 2 sin 1
n n n n . 5.17. ( ) = − + + − − 1 2 1 1 3 1 2 1
n n n n .
5.18. ( ) = + + − 1 3 1 1 1 n n n n . 5.19. ( ) = − 1 2 sin
1 n n n .
5.20. ( )
= − 1 3 3 sin
1 n n n n . 5. 21. ( ) = − − 1 1 6 cos 1
n n .
5.22. ( )
( ) = + + + − 1 1 1 1 2 1 n n n n n . 5.23. ( ) ( ) = − + − 1 1 ! 1 2 1 n n n n .
5.24. ( ) = + − − 1 2 4 2 1 2 1
n n n n . 5.25. ( ) = − 1 6 cos
1 n n n .
5.26. ( )
= + + − − 1 1 1 1 2 1 n n n n n . 5.27. ( ) ( ) = − + − 1 2 2 1 1 5 1 n n n n n .
5.28. ( ) ( ) = − + + − 1 1 1 1 ln 1 n n n n . 5.29. ( ) 1 1 n n tg n = − .
5.30.
( ) ( ) = + + − 1 2 1 ! 1 1 n n n n .
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