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#1 2012-06-11 13:53:36

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

Taylor polymomial/series of -ln(1-x)

I have to find the series for g(x) = -ln(1-x) using the function f(x) = 1/(1-x).
Seems simple as f(x) is just the derivative of g(x).

So I start by finding the series for 1/(1-x) =

k=0 to inf.

Now I am dont know where to go from here, I can integrate the whole series but that doesnt get me the right answer.

I also tried to work the series from ln(x) to -ln(1-x) but also got stuck.

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#2 2012-06-11 14:08:46

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

Re: Taylor polymomial/series of -ln(1-x)

What do you get when you integrate the series for 1/(1-x) ?


“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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#3 2012-06-11 19:09:23

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

Re: Taylor polymomial/series of -ln(1-x)

hi careless25,

Your integration method does work and your series for 1/(1-x) is correct.

So I have the same question as anonimnystefy, what did you get and why did you think it was wrong?

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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#4 2012-06-11 20:03:16

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

Re: Taylor polymomial/series of -ln(1-x)

Hi careless25;


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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#5 2012-06-12 05:03:10

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

Re: Taylor polymomial/series of -ln(1-x)

Yea I get your answer bobbym but wolfram gives another representation.
http://www.wolframalpha.com/input/?i=-l … esentation

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#6 2012-06-12 05:47:04

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

Re: Taylor polymomial/series of -ln(1-x)

If you mean this (see picture below), then it is the same.  There are 2k minus signs there and that will work out to be +

That's the problem with getting a computer to work it out ... you may not get the simplest form.

It happened to me recently with an integration involving trig.  Took 3 lines of working to change Wolfram's version into mine.

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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#7 2012-06-12 05:50:32

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

Re: Taylor polymomial/series of -ln(1-x)

Thanks! That explains a lot.

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#8 2012-06-12 06:49:17

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

Re: Taylor polymomial/series of -ln(1-x)

As Bob said,you will not always get the simplest form from a computer, but,  speaking in bobbym's name, more often you will. So don't discard computers right away. 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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#9 2012-06-12 08:13:47

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

Re: Taylor polymomial/series of -ln(1-x)

Hi;

Yea I get your answer bobbym but wolfram gives another representation.

Alpha's representation are often a little different. It does not reason the same way as we do. Different methods can sometimes produce the same answer in different forms.

But the series for 1 / ( 1 - x ) is a well know one in combinatorics so you will not need a computer. You can just integrate the general term.


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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#10 2012-06-12 08:41:55

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

Re: Taylor polymomial/series of -ln(1-x)

Ah,the most beautiful generating function.


“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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#11 2012-06-12 08:51:39

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

Re: Taylor polymomial/series of -ln(1-x)

Hi;

One last beef:

Although I applaud Wolfram for doing something for everyone to benefit by I have repeatedly emailed them about getting Alpha to understand Mathematica's syntax!


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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#12 2012-06-12 08:56:20

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

Re: Taylor polymomial/series of -ln(1-x)

You mean,so that you can use either?


“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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#13 2012-06-12 08:58:12

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

Re: Taylor polymomial/series of -ln(1-x)

Yes, Wolfram is a bit like a guy I knew who would use an English to English book to watch Benny Hill!


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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