Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

60, -10, 5/3, -5/18,

Answer 1

@amistre64

Answer 2

might help to see what the tops look like when they have like denominatos

Answer 3

so all have denominator 18?

Answer 4

1080, -180, 30, -5
------------------
18

well, the pattern seems to be wobbling back and forth between signs and getting smaller and smaller

Answer 5

ahhh I see so tht would be diverging?

Answer 6

and if miscalculated some of them :)

Answer 7

huh? :D

Answer 8

since they are bobbling back and forth, lets just absolute value it

1080, 180, 90, 5
---------------
18

can we define a expression for the top sequence?

Answer 9

since this is clearly not arithmetic ....dividing by 5 we get

{1080, 180, 90, 5}/5 = { 216, 36, 18, 1}

Answer 10

divide by 6: 36, 6 ,3 ,1/6

divide by 3: 12, 2 ,1 ,1/18

divide by 2: 6, 1 ,1/2 ,1/36

divide by 6: 1, 1/6 ,1/12 ,1/236

in some fashion or another the top is approaching 0

Answer 11

can you find a closed form to operate with?

Answer 12

60/6 = 10

10/6 = 5/3

(5/3)/6 = 5/18

its a geometric series

Answer 13

\[\sum_0~\frac{(-1)^n~60}{6^n}\]maybe

Answer 14

wait I am so lost!:(

Answer 15

this might be helpful

http://www.wolframalpha.com/input/?i=table+%28-1%29%5En*60%2F6%5En%2C+n%3D0+to+10

Answer 16

so it would be converging at 0?

Answer 17

the limit of the sequence is:

\[\lim_{n\to inf}~\frac{60~(-1)^{n+1}}{6^{n+1}}*\frac{6^{n}}{60~(-1)^{n}}\]
\[\lim_{n\to inf}~\frac{(-1)^{n+1-n}}{6^{n+1-n}}\]
\[\lim_{n\to inf}~\frac{(-1)}{6}\]
is that correct?

Answer 18

well that's not an option the answers are set up as

Diverges

Converges; 11100

Converges; 72

Converges; 0

Answer 19

lol, my latex coding is off .... the limit of the sequence goes to zero which indicates that the series has hope for convergence

Answer 20

Excellent! That's what I was thinking when I saw wolfram! :)

Answer 21

im reading something into it thats not there ... i was trying for convergent series for some odd reason :/

Answer 22

I see, :you were great either way :') thanks so much!!!

Answer 23

youre welcome

Answer 24

sequence diverges, because is not monotone

Answer 25

what??

Answer 26

sequence converges, when is monotone and has limit

Answer 27

so it can't be converges at 0??

Answer 28

\[a_n = 60(\frac{-1}{6})^n\]
\[\lim_{n\to \inf}|a_n| = 0\]

Answer 29

Yeah.. that seems accurate!

Answer 30

since |-1/6| < 1 .... it goes to zero

60*0 = 0

Answer 31

Yep! it'sdefinitely correct, thanks loads! @amistre64

Answer 32

youre welcome ... again :)