Question

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

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

this is a test question, and there was nothing in the lesson about convergence, divergence, or limits, so ill need an explanation please :(

Answer 1

can you recognize what type of a series this is ?

Answer 2

geometric because 10/60= 0.167 =r
and 5/3 / 10 = 0.167 =r

Answer 3

the consecutive terms maintain a fixed ratio

Answer 4

YES! so we can use the geometric series test. do you remember what that is ?

Answer 5

no :(

Answer 6

but i will gladly take notes

Answer 7

ok, so according to that test if the |r|<1, then your series converges. if it is >=1 then it diverges. since we know r=1/6 we know it converges.

Answer 8

know you can use the formula a1/1-r to find the sum or the limit it converges to.

Answer 9

a1 is your first term in the series.

Answer 10

what exactly is a sum, or limit? is it like an asymptote?

Answer 11

and what is the difference between a series that diverges and one that converges?

Answer 12

and is that formula, \[\frac{ a _{1} }{ 1-r }\] ?

Answer 13

yeah its like a value that your series is approaching.
a diverging series doesn't have a value that it converges to it goes to infinity whereas a converging series converges to a specific value.

Answer 14

yeah.

Answer 15

okay i just wanted to clarify. one sec.

Answer 16

60/ 1- (1/6)
60/(5/6)
= 72

Answer 17

so the series is converging because is ratios absolute value is less than one, and its limit is 72?

Answer 18

im pretty sure thats right based on what youve taught me

Answer 19

its convergent because the ratio is less than 1. the sum of this series is 72.

Answer 20

the sum of a geometric series is equal to it's limit. i think. let me check.

Answer 21

so i have one more problem that i have to solve that i dont know where to start with, would you be willing to keep helping me?

Answer 22

yup.

Answer 23

limit=sum in a geometric series.
and yes whats the question ?

Answer 24

A certain radioactive isotope has a half-life of 11 days. If one is to make a table showing the half-life decay of a sample of this isotope from 32 grams to 1 gram; list the time (in days, starting with t = 0) in the first column and the mass remaining (in grams) in the second column, which type of sequence is used in the first column and which type of sequence is used in the second column?

Answer 25

same test, based off the same crappy lesson, haha.

Answer 26

Lol.
alright can you make the table if i explain you the process and give you the value ?

Answer 27

probably

Answer 28

@macknojia

Answer 29

i got distracted my bad

Answer 30

alright so you know that the first value is 32 and then after 11 days its going to be 16 and then after the next 11 its going to be 8 and..

Answer 31

give me a set of values now which correlate to this relationship.

Answer 32

32/16=2=r
so the ratio is 2?

Answer 33

0-32
11-16
22-8
33-4
44-2
55-1

Answer 34

decreasing by a 1/2 and increasing by 11.
r=1/2. if the ratio was 2 this would be a diverging series.

Answer 35

okay

Answer 36

i think i get it

Answer 37

alright

Answer 38

Thank you @macknojia i had the same situation happen to me and your answers helped me ! :)