Question

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

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

Answer 1

ok, your common ratio is>?

Answer 2

Okay

Answer 3

your common ratio is not "okay" lol

Answer 4

do you know what a common ratio is?

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

Okay! I undertsand that!

Answer 10

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

Answer 11

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

Answer 12

oh, for a sequence to diverge and converge

Answer 13

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

Answer 14

confused**

Answer 15

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)

Answer 16

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
\(a_\infty=5/\infty=0\)

Answer 17

Ohhhhhh that makes sense okay

Answer 18

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)

Answer 19

What about if something diverges?

Answer 20

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)

Answer 21

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

Answer 22

0, since the number values get smaller and smaller

Answer 23

yes, that is correct

Answer 24

So that means that "sequence converges to zero"

Answer 25

^-^ Thanks a lot that makes sense!

Answer 26

(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))

Answer 27

And you asked about divergence, i will answer that now

Answer 28

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

Answer 29

Say you got
1, 2, 4, 8, 16, 32 .......

what is the pattern here, can you tell me?

Answer 30

the next number is being multiplied by 2

Answer 31

yes, r=2

Answer 32

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?)

Answer 33

Yes!

Answer 34

that means that sequence diverges.

Answer 35

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

Answer 36

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).

Answer 37

yes, what you said is correct

Answer 38

that is a good definition of convergence of a SEQUENCE.

Answer 39

Okay only for a sequence, got it.

Answer 40

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

Answer 41

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 ±∞)

Answer 42

is this making some sense?

Answer 43

I believe so, yes

Answer 44

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

Answer 45

The sequence converges to zero

Answer 46

yes

Answer 47

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

Answer 48

But in this case, the rule does not apply

Answer 49

this rule always applies
provided that ratio is: -1<r<1

Answer 50

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

Answer 51

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

Answer 52

lets see, but why does the common ratio r has to satisfy -1<r<1?
I will adress that now....

Answer 53

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

Answer 54

and then you get, roughly speaking

\(\large\color{black}{ \displaystyle a_\infty=5\cdot r^{\infty -1}}\)

Answer 55

Makes sense

Answer 56

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<r<1
then you are going to have \(\large\color{black}{ \displaystyle r^{\infty -1}=0}\)

Answer 57

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.

Answer 58

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,
\(\large\color{black}{ \displaystyle \sum_{ n=1 }^{ \infty } ~ A_n~=\frac{60}{1-{\rm -\dfrac{1}{6}}}=\frac{60}{\dfrac{7}{6}}=360/7 }\)

Answer 59

: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!

Answer 60

Anytime!