Question

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

108, -18, 3, -1/2, ...

I know it converges but how do I get the limit?

Answer 1

@BasketWeave

Answer 2

good, look up `infinite geometric series` formula in your notes

Answer 3

Well, you're dividing by (-6) each time, which basically means dividing by six and flipping the sign every time. First of all, can you see intuitively whether it's going to converge?

Answer 4

I'm not sure

Answer 5

Let me just confirm: are we looking for the convergence of the sequence, or for the series (i.e. the sum of all the terms in the sequence)?

Answer 6

the limit

Answer 7

Okay, suppose we were dividing by 6 every time, rather than (-6). Do you think the series will converge?

Answer 8

Oops, I meant "the sequence". Sorry!

Answer 9

as in get smaller?

Answer 10

Well, yes, they are getting smaller every time. "Converging" means that they approach some fixed number. What number do you think the sequence might approach in this case?

Answer 11

I was thinking 0 but we passed it...

Answer 12

Ah, I see why you're confused.

Answer 13

Try and find the next two or three terms in the sequence - that might help.

Answer 14

What would come after -1/2 ?

Answer 15

1/12

Answer 16

Great! And after that?

Answer 17

-1/72?

Answer 18

Yep! :)

Answer 19

So the size of the number keeps getting smaller, while the + or - sign gets changed every time.

Answer 20

So it does converge to zero, but it does it by kind of "bouncing" around on either side of zero. Like this:

Answer 21

so the limit is 0?

Answer 22

Yep!

Answer 23

Ooooooookay