Another puzzle about Fibonacci numbers

[c00ler]

Puzzle by hyh1048576:

Another puzzle about Fibonacci numbers(quite easy but interesting):
S=arccot(F(1))+arccot(F(3))+arccot(F(5))+arccot(F(
7))+......
Then S=? :?
Even a monkey can get the answer(using a calculator :) ).But prove it is more interesting.

Can anyone prove that S=Pi/2?

[Fusion_power]

Interesting in a way.

This brings up an unusual question. Pi is related to the distribution of prime numbers. The speed of light is related to the value of Pi.

What is the relationship of the speed of light to the distribution of prime numbers?

Fusion = specialist in off-the-wall thinking.

[mfgoode]

Interesting question c00ler and an interesting answer fusion power.
I understand what is implied in both question and answer provided both statements are correctly stated. Asfaik both are wrong.
No infinite series can give a correct value to pi. At most an approximation.
The speed of light is not related to pi as far as I know.
Ill try to prove the question or try to disprove it as a close appx,
If the speed of light is related to pi Ill br happy to learn about it
Mally
.

[xilman]

Originally Posted by Fusion_power

Interesting in a way.

This brings up an unusual question. Pi is related to the distribution of prime numbers. The speed of light is related to the value of Pi.

What is the relationship of the speed of light to the distribution of prime numbers?

Fusion = specialist in off-the-wall thinking.

Very little.

pi is a dimensionless quantity.

The speed of light has dimension length/time.

At first sight, the two are incommensurable. However, from the point of view of relativity (special and general) space-time is a four-dimensional manifold and the speed of light is but a scaling factor. In this view, metres/second is no more interesting than kilometres/mile or seconds/year.

So, from the point of view of relativity, the speed of light and pi are both dimensionless constants. There is a relationship for you.

I will be interested to hear of any relationship which, from the relativistic point of view, is other than as a scaling factor between two quantities measured in different but commensural units.

(To indicate that I'm aware of at least one more relationship between commensural units (more than one, in fact), I will utter the magic word \epsilon_0.)

Paul

[Citrix]

Could some one give an example of arcot?

What is arcot?

I understand that S=f(1)+f(3)+...+f(n). Is S=pi/2?

Citrix

[Numbers]

Citrix,
Arccot (with two c's) is the inverse cotangent.

Go here for more info:

http://mathworld.wolfram.com/InverseCotangent.html

[Citrix]

Isn't 1/tan X=cot X?
Then isn't arccot X=tanX?

Am I correct?

Citrix

[Wacky]

Originally Posted by Citrix

Am I correct?

Yes, and no, respectively.


In the first case, cotangent is the reciprocal of tangent.
However, arctangent is the inverse of tangent.

cot(x)*tan(x) =1
arctan(tan(x))=x

Using a simpler to understand function as an example:

x=1/(x^-1)
and
(x^(-n))^n = x

[Citrix]

Thankyou, My trignometery is rusty. Several years since I have needed to use it.

Citrix

[mfgoode]

Originally Posted by Wacky

Yes, and no, respectively.


In the first case, cotangent is the reciprocal of tangent.
However, arctangent is the inverse of tangent.

cot(x)*tan(x) =1
arctan(tan(x))=x

Using a simpler to understand function as an example:

x=1/(x^-1)
and
(x^(-n))^n = x

Arctan (tan(x)) = x is confusing.

We know that tan 45* = 1
For the reverse process Arc tan (1) =45*
Mally

[cheesehead]

Mally ,

I presume you meant:

Originally Posted by mfgoode

No

finite truncation of an

infinite series can give a correct value to pi. At most an approximation.

, so that when you later wrote

Ill try to prove the question or try to disprove it as a close appx,

, you meant

"... or try to disprove it as a finite truncation of some infinite series,"