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HSC 2014 MX2 Marathon ADVANCED (archive) (1 Viewer)

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Chlee1998

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Re: HSC 2014 4U Marathon - Advanced Level

what is the value of the infinite series, 1/1+ 1/2 +....+1/8+ 1/10 +...+1/18+ 1/20+.......
Btw this is the 1/1+1/2+1/3+....+1/n except without any terms with a 9 in the denominator
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

what is the value of the infinite series, 1/1+ 1/2 +....+1/8+ 1/10 +...+1/18+ 1/20+.......
Btw this is the 1/1+1/2+1/3+....+1/n except without any terms with a 9 in the denominator
So do you want a finite sum or infinite sum?
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

 

Chlee1998

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Re: HSC 2014 4U Marathon - Advanced Level

just to clarify, if the term has at least 1 digit which is 9, it is excluded from the series. So 1/998, 1/89 etc are all excluded
 

Chlee1998

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Re: HSC 2014 4U Marathon - Advanced Level

actually u can't find the exact value it converges to.. Change the question to 'explain why the series converges and roughly where it converges'
 

glittergal96

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Re: HSC 2014 4U Marathon - Advanced Level

Let be the set of reciprocals of k-digit positive integers that don't contain 9's.

Then the sum is



The monotone convergence theorem completes the proof of convergence.

To obtain an accurate estimate, exactly sum the contributions of the first few S_k and use the same GP to bound the error. (This takes a while though, because 9/10 is pretty close to 1.)
 
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Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

actually u can't find the exact value it converges to.. Change the question to 'explain why the series converges and roughly where it converges'
:(
 

Chlee1998

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Re: HSC 2014 4U Marathon - Advanced Level

In trapezium (quadrilateral with one side parallel) with sides XY and ZU parallel, Angle UXY=6 degrees, Angle XYZ= 42 degrees. Point B is one side XY such that angle XBU = 78 degrees and angle ZBY = 66 degrees. If XY and Zu are 1 cm apart (vertically) prove that XU- YZ +UB-ZB = 8cm
 

SilentWaters

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Re: HSC 2014 4U Marathon - Advanced Level

In trapezium (quadrilateral with one side parallel) with sides XY and ZU parallel, Angle UXY=6 degrees, Angle XYZ= 42 degrees. Point B is one side XY such that angle XBU = 78 degrees and angle ZBY = 66 degrees. If XY and Zu are 1 cm apart (vertically) prove that XU- YZ +UB-ZB = 8cm
Drop perpendiculars from and to . From a consideration of the right angled triangles formed thereby, it should be easy trigonometry that and Plug these into the left-hand side and you have your eight.
 

glittergal96

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Re: HSC 2014 4U Marathon - Advanced Level

It should suffice to prove that each component of the sum exceeds By the conditions, so any root of this where should be less than two. It follows that the reciprocal must exceed a half.
Why does the n-th root of a number bigger than two have to be less than two? This doesn't look right to me.
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

It should suffice to prove that each component of the sum exceeds By the conditions, so any root of this where should be less than two. It follows that the reciprocal must exceed a half.
At this part of your reasoning, are you saying that since:

because that does not seem to be a valid inference
 

glittergal96

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Re: HSC 2014 4U Marathon - Advanced Level



by AM-GM. (Noting that not all terms are equal, so we cannot have equality.)

Similarly, we have



Invert these inequalities and sum them to complete the proof.
 

SilentWaters

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Re: HSC 2014 4U Marathon - Advanced Level

woah what the hell am I doing, wishful thinking. sorry guys, ignore that
 

Axio

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Re: HSC 2014 4U Marathon - Advanced Level



by AM-GM. (Noting that not all terms are equal, so we cannot have equality.)

Similarly, we have



Invert these inequalities and sum them to complete the proof.
Noice. :D

Could you explain to me how you get that first line to fit the AM-GM inequality?
 

Chlee1998

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Re: HSC 2014 4U Marathon - Advanced Level

Drop perpendiculars from and to . From a consideration of the right angled triangles formed thereby, it should be easy trigonometry that and Plug these into the left-hand side and you have your eight.
how do you know that plugging those values into the LHS gives you 8 without using a calculator? You cannot use a calculator for this question
 
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