Question

determining convergent or divergent sequence

Answer 1

\[a_{n}= \frac{ 3^{n+1} }{ 5^{n} }\]

Answer 2

I would say divergent but I am not sure how to show it in a mathematical way

Answer 3

how do you determine if a sequence is div or cvg?

Answer 4

taking the limit

Answer 5

well take the limit then, what are waiting for?

Answer 6

you*

Answer 7

so that's all I would say\[\lim_{n \rightarrow \infty}\frac{ 3^{n+1} }{ 5^{n} }= \infty\]

Answer 8

how is that you found infinity?

Answer 9

you have infinity over infinity which is indeterminate remember!

Answer 10

but using l'hopital would just continue to yield an indeterminate

Answer 11

analytically the bottom goes to infinity really fast than the top
when try plugging some values you find the bottom is always in rush than the top
so that would mean the limit is zero

Answer 12

no using l'hopital would make more apparent that it is going to zero

Answer 13

note that \[\lim_{n\to \infty}r^n=0\]
exactly when \[|r|<1\]

Answer 14

so the sequence here cvg yes?

Answer 15

you could rewrite it as \[\lim_{n\to \infty}3\left(\frac{3}{5}\right)^n\]

Answer 16

3/5<1 so the the limit is zero

Answer 17

ok that makes it much clearer

Answer 18

can you explain to me the r^n statement you had from earlier though that was not covered during class for sequences but it was for series

Answer 19

remember you exponential when r<1 what is does
it goes to zero ( i mean the graph when we dealt with functions)
same applies to sequences

Answer 20

r=3/5 is just the ratio of the sequence
note that this sequence is geometric and we usually denote the ratio with an r

Answer 21

i should of explained this way
\[a^x,~~ when ~~|a|<1\]
the lim goes to zero when x grows large

Answer 22

recall this from calc1 and even precal :)

Answer 23

that makes sense thank you

Answer 24

no problem

Answer 25

another way is to show that \[a_n\] is monotonic and bounded
this one seems to be bounded for sure
and even monotonic
I'm saying because we know the behavior of exp function
so we embed that to sequence

Answer 26

if you are confused by monotonic stuff don't worry it is just a fancy way of saying it is
strictly increasing or strictly decreasing :)