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#1 2012-06-02 17:04:52

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Series Convergence

I am having trouble showing that the below series is convergent and finding its sum:


from n = 1 to inf.

I tried rewriting the series multiple ways but am not able to something i can use. Here is one I think I may be able to use:

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#2 2012-06-02 18:52:32

Bob
Administrator
Registered: 2010-06-20
Posts: 9,278

Re: Series Convergence

hi careless25

That re-arrangement works.  It makes both into geometric progressions.

First has 'r' = 1/4 with first term 'a' = 1

Second has r = 1/2 and a = 2

GPs are convergent if |r| < 1

and the sum to infinity is a/(1-r)

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2012-06-02 21:20:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

Hi;

The sum on the right you should recognize as 4.

The other sum,

you say,


Solving we get:


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#4 2012-06-03 03:59:03

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

Hi,

Thanks for your help bob and bobby!

bobbym: I am confused about why you multplied the sum by 1/4 and the steps onwards.

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#5 2012-06-03 04:08:35

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

Hi careless25

When you multiply the sum by 1/4 you get something that looks like your sum,but is smaller by 1.So:


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#6 2012-06-03 04:08:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

Hi;

The trick is useful in other types of problems especially generating functions and recurrences. I will not mention the first place people are exposed to it.

You are trying to get part of the above series in terms of Sn.  The boxed portion is 1 / 4 of Sn so we can replace it with 1 / 4 Sn. That leaves the equation to solve which gives us the sum.

It is amazing that we can change a sum into a linear equation!


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2012-06-03 06:13:26

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

Oh!
I know of the trick just that the way you wrote it out the first time made it seems like each proceeding step is equal to the previous one.

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#8 2012-06-03 08:12:52

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

I am stuck on another series, I just have to find if its convergent or divergent.


from n = 1 to inf.

I tried doing the Alternating Series Test(Leibniz Test) but I get upto:
lim as n--> inf


and dont know where to go from here.

EDIT: change of sign

Last edited by careless25 (2012-06-03 09:12:29)

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#9 2012-06-03 08:25:30

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

Hi careless25

Check this out:http://www.wolframalpha.com/input/?i=li … %29-n-1%29

It uses the idea of proving divergence using the ratio test.Click the "Show steps" button if you don't know how to get the limit and post here if you have questions. smile


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#10 2012-06-03 08:36:40

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

I am not sure if thats supposed to help as wolfram says ratio test is inconclusive:

http://www.wolframalpha.com/input/?i=su … D+1+to+inf

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#11 2012-06-03 08:43:28

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

Oh,I copied the problem incorrectly. Yes,the ratio turns out be 1/2 + i/2 or something like that.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#12 2012-06-03 08:46:51

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

I know what the ratio turns out to be, but I am wondering about how to show that the series diverges or converges. And how to calculate the ratio to show that.

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#13 2012-06-03 08:49:35

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

You cannot use the ratio test here. Did yo try any other convergence tests?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#14 2012-06-03 08:59:03

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

I got stuck using Alternating Series Test(aka Leibniz Test), cant do Ratio or Root Test and currently I am trying to see what happens to the partial sums of this series.

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#15 2012-06-03 09:05:23

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

Ok I rearranged the expression to this:

when I take the limit:



Last edited by careless25 (2012-06-03 09:10:37)

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#16 2012-06-03 09:10:12

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

Then your signs aren't right in post #8.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#17 2012-06-03 09:14:05

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

Thanks for pointing that out, but that still doesnt solve the problem. Fixed post #8
and my post above is still correct.

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#18 2012-06-03 09:24:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

Hi careless25;

For convergence a series has to first get smaller in absolute value of its terms. Does this one?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2012-06-03 09:43:58

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

faint, I have been at this for atleast a couple of hours.
Hmm so all i have to do is show that absolute value of a(n+1) > a(n) ????

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#20 2012-06-03 09:46:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

The tests you are using do not apply.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#21 2012-06-03 09:56:45

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

Yes I know but I am not sure which test would apply now...is this a geometric series? No. Can I use divergence test(n-th term test) on this? I dont think so because this is similar to the case of harmonic series combined with an alternating series(which converges even though the harmonic series diverges by itself).
With the comparision test or limit comparision test...I am not sure what to compare to, -1^n sqrt(n)?
Integral test wont work, this is not a p-series, AST, ASET, root and ratio all dont work.

Therefore by deduction LCT or Comparision Test should do it!

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#22 2012-06-03 10:02:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

The series diverges. Try to prove that.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#23 2012-06-03 11:24:23

careless25
Real Member
Registered: 2008-07-24
Posts: 560

Re: Series Convergence

I have given up on this for today, will look at it with a fresh mind tomo.

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#24 2012-06-03 16:13:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Series Convergence

Hi careless25;

Good idea! I am going to go eat.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#25 2012-06-04 06:48:22

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,048

Re: Series Convergence

Hi careless25

Last edited by anonimnystefy (2012-06-04 17:11:00)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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