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unicwaan
 one year ago
Determine whether the sequence converges or diverges. If it converges, give the limit.
60, 10, 5/3, 5/18, ...
unicwaan
 one year ago
Determine whether the sequence converges or diverges. If it converges, give the limit. 60, 10, 5/3, 5/18, ...

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SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1ok, your common ratio is>?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1your common ratio is not "okay" lol

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1do you know what a common ratio is?

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1XD Oh I thought you were telling me that, I'm sorry. I do not know what a common ratio is.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1common ratio, is a number by which you multiply to get to the next term each time.

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Oh okay so would it be (1/6)?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1So, if I had 1, 2, 4, 8, 16 .... then my common ratio is 2 (and common ratio is denoted as r, so in this example r=2)  Yes, r=1/6 is right

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Okay! I undertsand that!

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Note: This is a geometric series (because it is multiplied times some number (referred to as common ratio (r) ).

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Do you know what it means for a series to converge and for a series to diverge?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1oh, for a sequence to diverge and converge

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1No I do not, that's what I'm coversed on. My lesson never went over it :(

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1sequence, is just a list of terms. When you say that "sequence converges to 3" (for example 3) you are saying that as you take 1000th 100000th (and roughly speaking infinityth terms) they all will be approximately 3 (and each time they will get closer and closer to 3)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Say you started from 60, and you were to just divide by 1/6 (not 1/6). \(a_1=60\) \(a_2=10\) \(a_3=5/3\) \(a_4=5/18\) \(a_5=5/108\) and roughly speaking \(a_\infty=5/\infty=0\)

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Ohhhhhh that makes sense okay

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1No, you won't actually hit 0. Never! But you closer and closer approach to zero every time when you divide by 1/6. (this is the idea of a "limit" that you approach some value)

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1What about if something diverges?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1now, in our case we are multiplying (in the first post, should say multiplying too) by 1/6 so, all we change is that \(a_1=60\) \(a_2=10\) \(a_3=5/3\) \(a_4=5/18\) \(a_5=5/108\) and so on.... we still get \(a_\infty=\pm5/\infty=0\) (whether a positive or negative number is divided by infinity, you get 0 in either case)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1So the more terms you take, the close and close you approach what value?

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.10, since the number values get smaller and smaller

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, that is correct

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1So that means that "sequence converges to zero"

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1^^ Thanks a lot that makes sense!

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1(and that is the only case  i.e.  sequence converges to zero  for a series to converge. If you want we can talk about the convergence of a series as well (later on in this thread))

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1And you asked about divergence, i will answer that now

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1When I do reach series in my lesson I will come to you! and yes

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Say you got 1, 2, 4, 8, 16, 32 ....... what is the pattern here, can you tell me?

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1the next number is being multiplied by 2

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1And as you take more and more terms you multiply times 2 more and more.... so you are going to go into infinity. (makes sense?)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1that means that sequence diverges.

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Oh okay so diverges means the increase (or decrease) infinitely while converges ia to get closer and closer to a certain number?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1so sequence can diverge in 2 cases. (1) if you have an alternating sequence that is not approaching zero. Such that \(A_n=4(1)^n\), so your terms are going to be 4, 4, 4, 4, and so on.... and as n approaches infinity (n→∞) you don't know what your term is going to be because it can be either 4 or 4. (2) if your sequence goes into infinity (or negative infinity).

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, what you said is correct

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1that is a good definition of convergence of a SEQUENCE.

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Okay only for a sequence, got it.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1now, a series is the sum of terms in a sequence.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1think about it. Series is a sum of all terms in the sequence. So if sequence converges to 3 to 8 (or to anything besides 0) then you are going to be adding 3's 4's or something's forever and that will not have a defined sum (will go into ±∞)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1is this making some sense?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, so for example in your case, your sequence converges to what?

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1The sequence converges to zero

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1and the rule is that a geometric SERIES (a series that has all terms follow the pattern of multipling times some common ratio) will always converge if common ratio r is: 1>r>1

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1But in this case, the rule does not apply

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1this rule always applies provided that ratio is: 1<r<1

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1and so it is in your case, cuz r=1/6 so it is between 1 and 1

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1Oh okay, I was thinking 0 was r, but r is the common ratio which was 1/6, That makes sense

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1lets see, but why does the common ratio r has to satisfy 1<r<1? I will adress that now....

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1for any geometric sequence you multiply times r to get to the next term. So lets say you start from \(a_1=5\) (or from any nonzero first term) \(\large\color{black}{ \displaystyle a_1=5}\) \(\large\color{black}{ \displaystyle a_2=5\cdot r}\) \(\large\color{black}{ \displaystyle a_3=5\cdot r^2}\) \(\large\color{black}{ \displaystyle a_4=5\cdot r^3}\) so on.... \(\large\color{black}{ \displaystyle a_n=5\cdot r^{n1} }\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1and then you get, roughly speaking \(\large\color{black}{ \displaystyle a_\infty=5\cdot r^{\infty 1}}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1so, if r\(\ge\)1 or if r\(\le\)1 then the \(\large\color{black}{ \displaystyle r^{\infty 1}}\) part will go to either ∞ or ∞. but if 1<r<1 then you are going to have \(\large\color{black}{ \displaystyle r^{\infty 1}=0}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1your teacher will go over in detail, but the bottom line is that series is convergent, ONLY when sequence converges to 0. (there are many more tests for various series to converge/diverge that you will learn later in calc, but for geometric series, all you need to know is sequence converges to 0) and sequence convergence is when the \(a_\infty\) approaches one particular value.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1And for any geometric series \(\large\color{black}{ \displaystyle \sum_{ n=1 }^{ \infty } ~ A_n}\) the sum is given by \(\large\color{black}{ \displaystyle \sum_{ n=1 }^{ \infty } ~ A_n~=\frac{a_1}{1{\rm r}}}\) in your case, the terms 60 is \(a_1\) and r=1/6 so, \(\large\color{black}{ \displaystyle \sum_{ n=1 }^{ \infty } ~ A_n~=\frac{60}{1{\rm \dfrac{1}{6}}}=\frac{60}{\dfrac{7}{6}}=360/7 }\)

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.1:D Thank you for explaining so much to me! This really helped a lot. and I learn about that and summantion notation in my next lesson so thanks!
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