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#1 2014-02-15 22:34:16
- ffd
- Guest
Convergence of series
How can we found out of this series converges?
#2 2014-02-15 22:45:08
Re: Convergence of series
Nope, it diverges. IDK why
'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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#3 2014-02-15 22:54:02
- ffd
- Guest
Re: Convergence of series
I know it diverges. The question is to explain WHY it does that.
Anyone else?
#4 2014-02-16 01:20:42
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,621
Re: Convergence of series
hi ffd
Welcome to the forum.
Factorise the 5 and you have ∑k which is a well known divergent.
To prove it, you group 1 term, then the next 2 terms, then the next 4 terms then 8 terms ... and show that each group is greater than 0.5 thus an infinite number of 0.5s ..... that diverges.
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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#5 2014-02-16 01:37:41
Re: Convergence of series
Will you please do it? I have never known how to do these problems
'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 2014-02-16 01:43:35
- ffd
- Guest
Re: Convergence of series
Thanks bob, and thanks for an excellent answer. I will probably ask more questions here, I hope I will get more answers from you or answers from other people with the same quality.
#7 2014-02-16 03:06:56
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Convergence of series
Hi ffd;
Another interesting proof is by reductio ad absurdum.
Lets say it does converge.
then
adding the equal fractions pairwise
But this is
which is a contradiction and disproves the first statement, therefore the Harmonic series does not converge.
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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#8 2014-02-16 04:07:27
Re: Convergence of series
I still do not understand why this problem is related to the harmonic series
'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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#9 2014-02-16 04:08:39
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Convergence of series
Which problem?
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 2014-02-16 04:13:12
- wcy
- Member
- Registered: 2005-08-04
- Posts: 117
Re: Convergence of series
The first problem is essentially 5 times the harmonic series, hence, if the harmonic series diverges, so does the first series.
Last edited by wcy (2014-02-16 04:15:07)
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#11 2014-02-16 04:23:14
Re: Convergence of series
Now I see. thank you
'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 2014-02-16 04:25:20
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Convergence of series
There is a change of index but that is the Harmonic Series we are discussing, It is known to diverge since the time of Oresme, circa 1380.
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 2014-02-16 04:32:40
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,621
Re: Convergence of series
Will you please do it? I have never known how to do these problems
OK.
This has no finite sum, although if you construct a program to sum the terms, it gets slower and slower as you need approximately twice as many terms as before to add another 0.5 It's a useful reminder that infinity is quite large.
Bob
Last edited by Bob (2014-02-16 04:34:26)
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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#14 2014-02-16 04:43:01
Re: Convergence of series
thanks
'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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