Question

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

60, 10, 5/3, 5/18, ...

Answer 1

@perl

Answer 2

to go from first term to second term, you divide first term by 6.
to go from second term to third term, you divide second term by 6. , and the pattern continues

Answer 3

so lets make a rule for the n'th term

Answer 4

60*1/6\(^{n-1}\)

Answer 5

What is the difference between a converging and diverging series? That is where I am really getting stuck, and the lessons/textbook says nothing about it.

Answer 6

this is a sequence, not a series .
a series is a sum, a sequence is a list

Answer 7

ah, I keep getting them confused *facedesk*

Answer 8

a sequence converges if it gets closer and closer to a specific real number.
so for example this sequence converges to 2:

1.9, 1.99, 1.999, 1.9999, 1.9999, ...

Answer 9

ok does this sequence get closer and closer to a number. lets write the decimal equivalent of the sequence?
60, 10, 5/3, 5/18, ...
= 60 , 10, 3.33, 1.11, 0.37, 0.123, 0.041

Answer 10

It is getting closer and closer to 0

Answer 11

yes :)

Answer 12

So that is all?

Answer 13

a few more terms
60 , 10, 3.33, 1.11, 0.37, 0.123, 0.041, .0137, .00457, .0015, ..

Answer 14

this is not a mathematical proof, but it is evidence that it converges

Answer 15

if you want to be more rigorous, we can look at the limit of the nth term , defined by the rule or formula for the sequence

Answer 16

O_O that sounds complicated

Answer 17

you found
a(n) = 60 / 6^(n-1), where n=1,2,3...

Answer 18

what happens when n gets very large (goes to infinity)
you have
60 / 6^(very large integer) = 60 / very large ~ .000000... almost zero

Answer 19

ooohhh... that is less complicated than I thought :)

Answer 20

or you can even use infinity notation
60 / 6^(oo-1)
= 60 / (oo)
~ .000000... nearly zero

Answer 21

this is better argument to show it converges. you can even be more rigorous using an epsilon delta proof, which i will spare the details (i think its overkill here)

Answer 22

Thanks :D Could you help on one other?

Answer 23

ok