Seriously?

Calculus Level pending

If π \pi can be expressed as continued fraction, then what is the number which is surely less than or equal to the AM of all the numbers(except 1) in the continued fraction form?

Note: Give your answer to at least 2 decimals.


The answer is 2.68.

Kartik Sharma by Kartik Sharma , 21, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Kartik Sharma
Jun 8, 2014

This is just a Kitchinnie's constant problem

Largest number which is less than or equal to the AM of all numbers(AM-GM inequality)

Therefore, we get the answer as Kitchinnie's constant = 2.6853....

P.S. - Kitchinnie founded that for most real numbers x,

which can be expressed as

x = a 0 + a 0 + 1 a 1 + 1 a 2 + 1 a 3 + . . . . . { a }_{ 0 } + { a }_{ 0 }\quad +\quad \frac { 1 }{ { a }_{ 1 }+\frac { 1 }{ { a }_{ 2 }+\frac { 1 }{ { a }_{ 3 }\quad +..... } } }

( a 0 a 1 a 2 . . . . . . ) 1 / n ({ a }_{ 0 }{ a }_{ 1 }{ a }_{ 2 }......)^{ 1/n } as tend to infinity = Kitchinnie's constant = 2.68....

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You need to fix your question, and change "AM" to "GM", that is, not arithmetic mean, but geometrical mean.
K = ( ∏ i = 1 n a i ) 1 n K={ (\prod _{ i=1 }^{ n }{ { a }_{ i } } ) }^{ \frac { 1 }{ n } }

Michael Mendrin - 7 years ago

Log in to reply

But through AM-GM inequality,

AM is greater than or equal to GM

And that's what I have written, number which is surely less than or equal to AM.

Kartik Sharma - 7 years ago

Log in to reply

The question implicitly asks for the "largest number", since otherwise it would allow for infinitely many answers.

As Michael pointed out, you have only shown that the GM is bounded (by 2.68 in most cases), but have not established that the AM must also be bounded by 2.68. This seems somewhat unlikely.

How do you know that the answer cannot be 2.69 or anything higher?

Calvin Lin Staff - 7 years ago

Log in to reply

@Calvin Lin – Sir, I have edited the problem and now it says that "the number which is surely less than or equal to the AM." I hope now this is correct and if not, I will make it GM. Please reply me on this context!

Kartik Sharma - 7 years ago

I checked out google there is nothing called Kitchinnie's constant.

@Calvin Lin Can you make a note on Kitchinnie's constant explaining it in depth?

Pinak Wadikar - 6 years, 11 months ago

Log in to reply

Pinak Wadikar Actually,he has misspelled it.The correct name is Khinchin‘s Constant \textbf{Khinchin`s Constant} .You can read about it here

Abdur Rehman Zahid - 6 years, 6 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...