Question

Can someone help me determine whether the sequence converges or diverges. If it converges, give the limit.

108, -54, 27, -27/2,........


Thank you!!!!

Answer 1

you would have to determine if there is some formula that can be developed to address the pattern

Answer 2

the last 2 terms might give us a clue

Answer 3

what you mean?

Answer 4

would it diverges?

Answer 5

i mean, that in order to determine what it does, it would be nice to see how the sequence is being formed.

the last 2 terms that are presented give us a clue to what they are doing to form the sequence.

Answer 6

what is the most reasonable explanation on how we get from:

27 to -27/2 ??

Answer 7

dividing?

Answer 8

yes, specifically ... dividing by -2

Answer 9

does this idea hold for the other terms as well?

can we get from one term to the next by dividing it by -2?

Answer 10

yes, it works for the others also

Answer 11

the next term would be 27/4?

Answer 12

then we can assume this is a geometric sequence:\[a_n=-\frac{1}{2}a_{n-1}\]
yes, 27/4 would be the next term :)

Answer 13

so how do we find if it diverges or converges?

Answer 14

there are a variety of methods available to us, i would prolly use a ratio test ... or, if a less stringent proof is required; 27/2^n gets very very small as n gets ver very big

Answer 15

how do I use the ratio test?

Answer 16

notice that out pattern is basically\[a_n=\frac{27}{2^n}\] for some starting integer value of n

Answer 17

the ration a{n+1}/a{n} is the ratio test

Answer 18

so the n I would plug in a number?

Answer 19

\[\frac{27}{2^{n+1}}\div \frac{27}{2^n}\]
\[\frac{\cancel{27}}{2^{n+1}} \frac{2^n}{\cancel{27}}\]
\[\frac{2^n}{2^{n+1}}=\frac12 \]
\[\frac{\cancel2^n}{\cancel2^{n}~2^1}=\frac12 \]

Answer 20

A general test would be to see if your ratio has an absolute value of less than 1 (-1 <r< 1). If this is true then your series converges, that is as we get farther into the sequence the terms get closer and closer to zero.

Answer 21

yes, the sequence tends to zero .... not too sure what i did there tho lol

Answer 22

lol so the series would converges at 0?

Answer 23

Yes. Despite the flip/flop of +/-, the terms will get smaller and smaller until, at least on your calculator, a term will be zero.

Answer 24

ohh, I got it now. Thank you so much @amistre64 and @mrbarry :)

Answer 25

You are very welcome. Math teachers DO want you to understand.

Answer 26

how can I give you both medals? true, and I understand it now. Thank you again.

Answer 27

i dont need a medal, i have enough of the useless trinkets floating about .... barry deserves to be leveled up tho

Answer 28

oh, kk then and how do I give him a medal then? and thank you.