Question

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

Answer 1

this is a converge correct?

Answer 2

why

Answer 3

btw, you're correct! it converges... can you explain why it converges ? :)

Answer 4

because it is infinate?

Answer 5

infinite *

Answer 6

not all infinite sequences converge

Answer 7

here is the rule :
An infinite geometric sequence converges if \(\large |r| < 1\)

Answer 8

whats the common ratio of the given sequence ?

Answer 9

-6?

Answer 10

@ganeshie8

Answer 11

common ratio = (next term) / (present term)
= -18/108
= -1/6

Answer 12

\(\large |r| = |-1/6| < 1\)
So the given sequence converges.

Answer 13

oh oops i thought i was looking for which number it was divided by

Answer 14

yes you're close :)

Answer 15

and for the limit, how do i get that? its confusing for me :/

Answer 16

Notice that the terms are decreasing..

Answer 17

what do you think the 100th will be ?

Answer 18

it will be close to 0, right ?

Answer 19

ohhh

Answer 20

108, -18, 3, -1/2, .......... 0, 0, 0...

Answer 21

so 0! (:

Answer 22

after n= 100 or so, the remaining terms will be close to 0
so the limit of the sequence is 0.

Answer 23

sorry if im asking so many questions, im doing a pre test and at the end it does not say which ones i get wrong so i want to se if i understand them before i move on the the actual test

Answer 24

nice :) when a geometric sequence converges,
we can find the corresponding infinite series sum also.... do u know the infinite series sum formula ?

Answer 25

\[\large S_{\infty} = \dfrac{a_1}{1-r}\]

Answer 26

seen this before ? :)

Answer 27

yes, its in my book(:

Answer 28

thank you!

Answer 29

good, plug the values to find the infinite series sum :

\[\large S_{\infty} = \dfrac{108}{1 - (-1/6)} = 648/7\]

Answer 30

so the sequence converges to 0
and the series converges to 648/7