Thread: series and sequence help

sorry for the image but i am having trouble proving the answers i get while doing my first problem



(a) I know the sequence converges to 0 since if you take the limit as n goes to infinity, then a(n) will go to 0 because fractions multiplied by themselves many times that are below 1 go to 0. but how do i prove this answer??

(b) is this geometric?

(c) does it converge because its a constant ratio by geometric series test?

(d) does it converge to 9?

(e) i know that the answer is yes, .999... is equal to 1. but how do i prove that? would i set up a series of some kind??

thanks

Originally Posted by Arez

(b) is this geometric?

For your series

It is geometric if

Originally Posted by Arez

(c) does it converge because its a constant ratio by geometric series test?

Only if

(e) Let

then

subtract x from both sides, x from the left and from the right

so or x=1.

(b) geometric with r=.9, Geo's converge when -1<r<1.
The ratio test is basically a limit comparison to a geometric.
The limit of adjacent terms is the search for the essential r.

(c) and (d) yes it converges to 9...

the limit is

(a) as n goes to infinity

Yes, the original poster said he knew that in his original post.