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#1 2015-08-15 00:46:04

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Log(2)

How do I find a closed form for the following:

1/1 - 1/2 + 1/3 - 1/4 + ...


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#2 2015-08-15 01:59:49

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

Re: Log(2)

There are many ways to try. What have you tried first?


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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#3 2015-08-15 02:00:44

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Log(2)

The answer as M suggests is Log(2)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2015-08-15 02:04:43

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

Re: Log(2)

That is correct and what else?


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 2015-08-15 02:20:33

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Log(2)

I searched the internet but did not undrstand it.

Is there a generating function way?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2015-08-15 02:22:30

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

Re: Log(2)

I do not remember offhand.


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 2015-08-15 02:24:52

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Log(2)

Okay, how do I do it?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2015-08-15 02:26:44

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

Re: Log(2)

I have taught you how to do a Fourier series? Did you try that?

http://math.stackexchange.com/questions … -frac-1k1k

Simplest to me is to remember the power series expansion of log(1+x).

Series[Log[1 + x], {x, 0, 10}]

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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#9 2015-08-15 02:45:05

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Log(2)

This is a Taylor Series.

What you are doing is a proof. You can only do this when you know the answer.

What if I am not given the answer to begin with?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#10 2015-08-15 02:49:57

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

Re: Log(2)

That is the Taylor series so I just substitute x = 1. But first check about convergence as power series have restrictions, that is why I sent you to that link.


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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#11 2015-08-15 02:51:34

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Log(2)

I already knew this proof. I am looking for something else


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2015-08-15 02:55:36

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

Re: Log(2)

The SE did not satisfy you? Then you must use EM! But according to that Patrick fellow we'all aint doing math when we do that... I say kaboobly doo!

Someone is serially upvoting me at the SE. They will only be removed if they exceed a certain limit from one person in a given amount of time.


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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#13 2015-08-16 03:44:08

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Log(2)

There is no cute proof? Not even one using a gf?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#14 2015-08-16 06:24:54

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: Log(2)

hi Agnishom

I don't see why you are against the log series method.  There's plenty of mathematical 'solutions' that require you to 'know' the answer. 

eg. 

Do you solve this from first principles or do you say ... I know that

so I'm going to make an educated guess that the answer is

and I'll check to see if I've got it right.

eg.  A rectangle has length 4 longer than its width and an area of 21.  What is the width ?

I'll pretend I know the answer and I'll call it x.

That makes the length x+4 and the area x(x+4)

So x could be -7 or 3.  I don't think it can be -7 as lengths are not usually negative so I'll try x = 3

=> length = 7 => area = 21.

My method led to a wrong answer which I was able to discard and also to an answer that fits.  So I'll assume I've solved the problem.  (Although, strictly speaking, I still need to justify that there are no other solutions smile )

So say, this expansion looks a bit like a log expansion.

Then I happen to notice that putting x = 1 gives the given series => this is log(2).

Note that for |x|  ≤  1 log(2) = the series .... that equals means that any other answer you might find by another method must also = log(2), so problem solved.

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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#15 2015-08-16 09:18:23

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

Re: Log(2)

There is no cute proof? Not even one using a gf?

I think the power series done by Bob and I is close to that.

A "cute way" would be to determine the constant to as many digits as we can and using a PSLQ on it.


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.

Offline

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