Question

please... someone help .-.

Answer 1

Geometric sequences

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

60, -10, five divided by three, negative five divided by eighteen, ...

Answer 2

What kinds of tests have been discussed in your classes?

Answer 3

what do you mean by tests?

Answer 4

In answering in that way you've told me. ;) Can you write the general term for this sequence?

Answer 5

no because i dont even know what the pattern is :/

Answer 6

I'm sure you can get it!
- What happens to the signs?
- And, forgetting about the signs of the terms, what happens to their size as you go from one to the next?

Answer 7

The second question is the important one where convergence is concerned.

Answer 8

is it divided by negative 6?

Answer 9

That's right! The test I was referring to is this : If each new term in a sequence is obtained by multiplying the previous term by a number whose _absolute value_ is less than 1 then the sequence converges. In this case :\[\left| -\frac{ 1 }{ 6 } \right| < 1\] and the sequence converges.

Answer 10

Now we want the limit. We can only get an absolute value for the limit because the sign alternatives, right? So let's forget about the sign. The general term is \[60(\frac{ 1 }{ 6^{n} })\]
You can see that this works for n = 0, 1, 2, 3, ... . I don't know what theory you have at your disposal for calculating limits?

Answer 11

wait how do i tell whether it is converges and diverges?

Answer 12

It converges because to go from one term to another you multiply by -1/6 and this is less than one in absolute value. (If this multiplier were more than one then the sequence would be divergent.)

Answer 13

that makes sense but how do you get to a limit?

Answer 14

I can tell you how most of us would do it. As n becomes large 6^n becomes very large and, hence, 1/6^n becomes very small, vanishing in the limit. This amounts to saying that the limit of this function is zero.

Answer 15

wait so the bigger the number the larger the limit?

Answer 16

Notice that the big, big number is in the _denominator_. If it had been in the numerator then, roughly speaking, the sequence might diverge. This is actually a big topic in mathematics that has kept some of the greatest minds quite busy. Fortunately in the case of geometric sequences the rule for convergence is simple. Just look at the multiplier and compare its absolute value to one.

The _smaller_ the multiplier the _faster_ the sequence will converge. The multiplier will be smaller if the number in its denominator is big.

Answer 17

that makes sense c: thank you

Answer 18

You're welcome.