Question

How do you know if a sequence has a limit?

Determine whether the sequence converges or diverges. If it converges, give the limit.
60, 10, 5/3, 5/18...

Answer 1

looks geometric as each term is the previous one divided by 6

Answer 2

Yeah, that's the name of the lesson, geometric sequences...
But I don't know how to find the limit to know if it is convergent or divergent.

Answer 3

it does say "sequence" right, not "series"
the denominator gets larger, the terms go to zero

Answer 4

Yes, sequences. So if the terms go to zero, but never reach it, does it not have a limit?

Answer 5

the limit is zero

Answer 6

Your first term (a) is 60. Your common ratio (r) is (1/6). Thus, a formula for the nth term of your sequence is \[a _{n}=60*(\frac{ 1 }{ 6 })^{n-1}\] for n=1, 2, 3, 4, and so on. Note that if n=1, we get a1 = 60, as expected.

So the problem boils down to whether or not \[(\frac{ 1 }{ 6 })^{n-1}\] converges or not.
What do you think it does? 1, 1/6, 1/36, 1/216, ...

Answer 7

I don't know.. :( Converges to whattttt?

Answer 8

I don't even know what converging and diverging means.. D:

Answer 9

@whpalmer4

Answer 10

Hello, BG!
Have you any way of looking up "convergent sequence"?

Your '60, 10, 5/3, 5/18... " is a sequence. As you can see for yourself, each term of this sequence is smaller than the last one. And if you look carefully, you may see that the 2nd term was obtained by multiplying the first term by the fraction, 1/6.

We'd call the first term (60) "a" and that fraction (1/6) " r ", where
a=first term of the sequence, and
r = the common ratio.

Answer 11

Before we do anything more, let's see if our common ratio, r, r=1/6, is correct.

Multiply: 60(1/6)=10? Yes.
Multiply: 10(1/6) = 10/6 = 5/3? yes.
Multiply: (5/3)(1/6) =5/18? yes

So our common ratio, 1/6, is correct.

If you look carefully at the terms of this sequence, you'll see that every term is smaller than the previous term, by a factor of 1/6.

In the end, your terms will approach zero.

Would that mean your sequence is
(1) convergent, or
(2) divergent?

Answer 12

Convergent?

Answer 13

@mathmale

Answer 14

Yes, it is convergent.

Answer 15

This series converges to 0 because for any number you might choose for "how close" the series should get to 0, we can find a term of the series which is closer, if we just go far enough along the list of terms. Does that make sense?

Answer 16

I think sooo...