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#1 2011-05-03 21:00:16
- killbox
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- Registered: 2011-05-03
- Posts: 3
Show that Cn is a decreasing sequence of positive numbers?
show that Cn is a decreasing sequence of positive numbers
Cn = 1 + 1/2 + ... + 1/n - ln(n)
where n is part of N excluding 0
Thanks! just can't find the steps to prove it
(2nd year university maths)
Last edited by killbox (2011-05-03 21:01:37)
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#2 2011-05-03 21:30:41
- 123ronnie321
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- Registered: 2010-09-28
- Posts: 128
Re: Show that Cn is a decreasing sequence of positive numbers?
Hi killbox,
I shall show you one of the approach.
Plot a rough graph of y = 1/x. Now you know C(1) = 1. C(2) = 1 + 1/2 - Log2. Note that Log 2 is the area of the graph 1/x from x = 1 to x = 2. and 1/2 is the area under y = 1/2 from x = 1 to x = 2. Thus Log2 > 1/2. Second term is less than 1. Apply similar approach for C(3)
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#3 2011-05-03 23:15:13
- 123ronnie321
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Re: Show that Cn is a decreasing sequence of positive numbers?
Hi Killbox,
Can you show that C(n) will always be positive?
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#4 2011-05-03 23:39:23
- killbox
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Re: Show that Cn is a decreasing sequence of positive numbers?
not too sure about that at the moment.
thanks for your help 
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#5 2011-05-03 23:47:08
- 123ronnie321
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- Registered: 2010-09-28
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Re: Show that Cn is a decreasing sequence of positive numbers?
You are welcome!
Did you understand the first part?
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#6 2011-05-03 23:48:24
- killbox
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- Registered: 2011-05-03
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Re: Show that Cn is a decreasing sequence of positive numbers?
yeah.!
do we just say it is positive from the graph?
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#7 2011-05-03 23:56:20
- 123ronnie321
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- Registered: 2010-09-28
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Re: Show that Cn is a decreasing sequence of positive numbers?
It think it is possible. But you will have to prove it logically. Try it. Try to see the behavior of C(n) graphically for n = 2,3 and then a bit larger.
Come back if you have any problem.
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#8 2011-05-04 00:02:44
- gAr
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- Registered: 2011-01-09
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Re: Show that Cn is a decreasing sequence of positive numbers?
Hi all,
The same question is discussed here: http://uk.answers.yahoo.com/question/index?qid=20110503043025AAaeWnn
A good solution, which I liked is:
Last edited by gAr (2011-05-04 00:04:00)
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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#9 2011-05-04 00:09:02
- 123ronnie321
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Re: Show that Cn is a decreasing sequence of positive numbers?
Hi gAr,
nice solution!
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#10 2011-05-04 00:14:28
- gAr
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Re: Show that Cn is a decreasing sequence of positive numbers?
Hi 123ronnie321,
Credits to the person who answered in that forum!
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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#11 2011-05-05 03:52:16
- 123ronnie321
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- Registered: 2010-09-28
- Posts: 128
Re: Show that Cn is a decreasing sequence of positive numbers?
Hi Killbox,
I stumbled upon this while searching the net http://math2.org/math/constants/gamma.htm
Last edited by 123ronnie321 (2011-05-05 03:56:22)
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