Determine whether the sequence converges or diverges. If it converges, give the limit.
60, -10, 5/3, -5/18, ...

- unicwaan

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- SolomonZelman

ok, your common ratio is>?

- unicwaan

Okay

- SolomonZelman

your common ratio is not "okay" lol

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## More answers

- SolomonZelman

do you know what a common ratio is?

- unicwaan

XD Oh I thought you were telling me that, I'm sorry. I do not know what a common ratio is.

- SolomonZelman

common ratio, is a number by which you multiply to get to the next term each time.

- unicwaan

Oh okay so would it be (-1/6)?

- SolomonZelman

So, 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

Okay! I undertsand that!

- SolomonZelman

Note:
This is a geometric series (because it is multiplied times some number (referred to as common ratio (r) ).

- SolomonZelman

Do you know what it means for a series to converge and for a series to diverge?

- SolomonZelman

oh, for a sequence to diverge and converge

- unicwaan

No I do not, that's what I'm coversed on. My lesson never went over it :(

- unicwaan

confused**

- SolomonZelman

sequence, 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 infinity-th terms) they all will be approximately 3 (and each time they will get closer and closer to 3)

- SolomonZelman

Say 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

Ohhhhhh that makes sense okay

- SolomonZelman

No, 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

What about if something diverges?

- SolomonZelman

now, 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

So the more terms you take, the close and close you approach what value?

- unicwaan

0, since the number values get smaller and smaller

- SolomonZelman

yes, that is correct

- SolomonZelman

So that means that "sequence converges to zero"

- unicwaan

^-^ Thanks a lot that makes sense!

- SolomonZelman

(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

And you asked about divergence, i will answer that now

- unicwaan

When I do reach series in my lesson I will come to you! and yes

- SolomonZelman

Say you got
1, 2, 4, 8, 16, 32 .......
what is the pattern here, can you tell me?

- unicwaan

the next number is being multiplied by 2

- SolomonZelman

yes, r=2

- SolomonZelman

And as you take more and more terms you multiply times 2 more and more.... so you are going to go into infinity. (makes sense?)

- unicwaan

Yes!

- SolomonZelman

that means that sequence diverges.

- unicwaan

Oh okay so diverges means the increase (or decrease) infinitely while converges ia to get closer and closer to a certain number?

- SolomonZelman

so 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

yes, what you said is correct

- SolomonZelman

that is a good definition of convergence of a SEQUENCE.

- unicwaan

Okay only for a sequence, got it.

- SolomonZelman

now, a series is the sum of terms in a sequence.

- SolomonZelman

think 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

is this making some sense?

- unicwaan

I believe so, yes

- SolomonZelman

yes, so for example in your case, your sequence converges to what?

- unicwaan

The sequence converges to zero

- SolomonZelman

yes

- SolomonZelman

and 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

But in this case, the rule does not apply

- SolomonZelman

this rule always applies
provided that ratio is: -1

- SolomonZelman

and so it is in your case, cuz r=-1/6
so it is between -1 and 1

- unicwaan

Oh okay, I was thinking 0 was r, but r is the common ratio which was -1/6, That makes sense

- SolomonZelman

lets see, but why does the common ratio r has to satisfy -1

- SolomonZelman

for 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 non-zero 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....
\(\large\color{black}{ \displaystyle a_n=5\cdot r^{n-1} }\)

- SolomonZelman

and then you get, roughly speaking
\(\large\color{black}{ \displaystyle a_\infty=5\cdot r^{\infty -1}}\)

- unicwaan

Makes sense

- SolomonZelman

so, 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

- SolomonZelman

your 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

And 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

: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!

- SolomonZelman

Anytime!

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