anonymous
  • anonymous
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 :(
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
can you recognize what type of a series this is ?
anonymous
  • anonymous
geometric because 10/60= 0.167 =r and 5/3 / 10 = 0.167 =r
anonymous
  • anonymous
the consecutive terms maintain a fixed ratio

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anonymous
  • anonymous
YES! so we can use the geometric series test. do you remember what that is ?
anonymous
  • anonymous
no :(
anonymous
  • anonymous
but i will gladly take notes
anonymous
  • anonymous
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.
anonymous
  • anonymous
know you can use the formula a1/1-r to find the sum or the limit it converges to.
anonymous
  • anonymous
a1 is your first term in the series.
anonymous
  • anonymous
what exactly is a sum, or limit? is it like an asymptote?
anonymous
  • anonymous
and what is the difference between a series that diverges and one that converges?
anonymous
  • anonymous
and is that formula, \[\frac{ a _{1} }{ 1-r }\] ?
anonymous
  • anonymous
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.
anonymous
  • anonymous
yeah.
anonymous
  • anonymous
okay i just wanted to clarify. one sec.
anonymous
  • anonymous
60/ 1- (1/6) 60/(5/6) = 72
anonymous
  • anonymous
so the series is converging because is ratios absolute value is less than one, and its limit is 72?
anonymous
  • anonymous
im pretty sure thats right based on what youve taught me
anonymous
  • anonymous
its convergent because the ratio is less than 1. the sum of this series is 72.
anonymous
  • anonymous
the sum of a geometric series is equal to it's limit. i think. let me check.
anonymous
  • anonymous
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?
anonymous
  • anonymous
yup.
anonymous
  • anonymous
limit=sum in a geometric series. and yes whats the question ?
anonymous
  • anonymous
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?
anonymous
  • anonymous
same test, based off the same crappy lesson, haha.
anonymous
  • anonymous
Lol. alright can you make the table if i explain you the process and give you the value ?
anonymous
  • anonymous
probably
anonymous
  • anonymous
@macknojia
anonymous
  • anonymous
i got distracted my bad
anonymous
  • anonymous
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..
anonymous
  • anonymous
give me a set of values now which correlate to this relationship.
anonymous
  • anonymous
32/16=2=r so the ratio is 2?
anonymous
  • anonymous
0-32 11-16 22-8 33-4 44-2 55-1
anonymous
  • anonymous
decreasing by a 1/2 and increasing by 11. r=1/2. if the ratio was 2 this would be a diverging series.
anonymous
  • anonymous
okay
anonymous
  • anonymous
i think i get it
anonymous
  • anonymous
alright
anonymous
  • anonymous
Thank you @macknojia i had the same situation happen to me and your answers helped me ! :)

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