Question

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

4, 12, 36, 108,

Answer 1

converges i think your multypliyng 3

Answer 2

Converges; 484

Diverges

Converges; 52

Converges; 160

Answer 3

are the answers they give me

Answer 4

Well, this can be written as:

\[\sum_{i=1}^{\infty} 4(3)^{n}\]

Answer 5

thanks

Answer 6

This is a geo-series of the form ar^n. When the absolute value of r is greater than or equal to 1, the series diverges. So this is a divergent series.

Answer 7

where did you get the equation

Answer 8

o got the answer to be (c) is the right

Answer 9

i got the answer to be (c) is this right

Answer 10

It is a general form of a geometric series. If you can put the series into a form of ar^n, you can write it in the way I did.

Answer 11

ahh ok well thank you and yes i do believe that c is right

Answer 12

so is my answer right???

Answer 13

yes its right

Answer 14

I would say diverges, but oh well.

Answer 15

no when i just did the equation it put me into the converges answer of c

Answer 16

okay so (c) is the right answer

Answer 17

Can you show me how then?

Answer 18

I would like to see your reasoning.

Answer 19

So would I

Answer 20

yea if i can find the paper i did it on i will show but i want to see @TNGs reasoning

Answer 21

i dont really have a reasoning i just use the eqaution and i dont really know how to use this software yet to show you all my work in a decent way

Answer 22

There is no work needed with this problem. There is a general form with specific conditions for convergence or divergence.

\[\sum_{i=1}^{\infty} ar ^{n}\]

If the absolute value of r is greater than or equal to 1, the series diverges. In this case, your a is 4 and your r is 3. This equation represents that for every subsequent part of the series, you are multiply 4 by an increasingly higher power of 3. Regardless of the formula and its conditions, think of it. When n = 1 you have 4, n = 2 you have 36, n = 3 you have 108, n = 4 you have 324, n = 5 you have 972. This number will get infinitely higher.

Answer 23

ok i stand corrected i messed up in the equation sorry it was a fools mistake

Answer 24

jsut hop in so is this problem a diverges

Answer 25

This diverges, correct.

Answer 26

Now just for the sake of finishing the explanation of a geometric series I would want to show the conditions for convergence and how you find the limit. With permission of course.

Answer 27

go right ahead

Answer 28

Alrighty.
Now, if the absolute value of r is between 0 and 1, the series converges. This is the formula to find out what numberit converges to.

\[\frac{ a }{ 1-r }\]

For example, say you had the series of 108, 36, 12, 4, 4/3......this would represent a series in the form of

\[\sum_{i=1}^{\infty} 108(\frac{ 1 }{ 3 })^{n}\]

This means my a = 108 and my r = 1/3. Therefore this fits the conditions of convergence and the limit would be

\[\frac{ 108 }{ 1-\frac{ 1 }{ 3 } }\]
\[\frac{ 108 }{ \frac{ 2 }{ 3 } } = 162\]