1549901369-Elements_of_Real_Analysis__Denlinger_

(^474) Chapter 8 • Infinite Series of Real Numbers
8.
v'2 v'4 v'6 JS JI5
3. 5 + 5. 7 + 7. 9 + 9. 11 + 11. 13 + ...
1 1·2 1 ·2· 3 1 ·2·3·4

9. 3 + 3.5 +~ + 3 .5.7.9 +···

1 2 3 4 5

-~~+ + + + +···

~ ~ ff.D #5-6 vim

10.

1 4 27 64 3, 125

-+-+-+-+--+···

e e^2 e^3 e^4 e^5

11.

12.

1 1 1 1

ln2 + (ln3)^2 + (ln4)3 + (ln5)4 + · · ·

In Exercises 13 - 30, use tests given in Sections 8.1 and 8.2 to determine whether
the series converges or diverges.

13. ~~

L..., n^2 - 5

n=l

~lnn

15. L...,- 2
    n=2 n


16. f ln~
    n=2 n

19. f^1 + cosn

n=l sin

2

n

00

21. '"'!!'.._
    L..., en
    n=l
    oo I


22. '"'!!:.: L..., en
    n=l


23. ~sinn(~)

oo ( )2n

27. ~ n3~ 1

00 1

14. '"' L..., -----,,3n 2 --- lOn

n=l

16. f lnn

n=2 5n

18. f fa
    n= 2 lnn
    20 ~^7 fa
    · ~ n^2 + 6ifri
    00 3


19. '"'!2:_
    L..., 3n
    n=l


20. f ln(~3)
    n
    n=2


21. oo (3n + 5)n/ 2
    2= 2n+1
    n=l


22. ~cosn (n
    2
    :

1

)

1 1 1 1 1 1 1 1

~.l+~+~+~+w+~+w+~+~+···

1 1 1 1 1 1 1 1 1

30. 22 + 1 + 42 + 32 + 62 + 52 + 32 + 72 + 102 + 92 + ...

00 1

31. Prove that ~ n(ln n)P converges if and only if p > 1.