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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hi careless25

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

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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