Prove the sum of this series =pi/3

In summary: I'm like Seraph, I just don't post unless someone else does.In summary, the series 1 + \frac{1}{5} - \frac{1}{7} - \frac{1}{11} + \frac{1}{13} + \frac{1}{17} - ... can be proven to equal \frac{\pi}{3} through various methods, including using sum notation and arctan. However, the poster "Cheeky CompuChip" chooses not to reveal the solution and instead teases others to figure it out themselves. The concept of transcendental numbers is also discussed and it is mentioned that \pi is a transcendental number. The
  • #1
santa
18
0
prove the sum of this series =pi/3

[tex]1 +\frac{ 1}{5} -\frac{ 1}{7} - \frac{1}{11 }+\frac{1}{13} + \frac{1}{17} - ...[/tex]
 
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  • #2
Surely you must have some idea.
Why don't you start by writing it out in sum notation ([tex]\sum_{n = 0}^\infty \cdots[/tex])?
 
  • #3
Cheeky CompuChip :P You know he knows the solution, he just wants to tease us :P Look at the rest of his posts to see what I mean.

I've got a nice solution for this though =] I'll post it when someone else asks for it, because I want to let some other guys give this a try. It's not actually that hard, this one.
 
  • #4
a twisted Gregorian pi series , missing fractions of odd multiples of 3, is all what it is.
Gib Z said:
Cheeky CompuChip :P You know he knows the solution, he just wants to tease us :P Look at the rest of his posts to see what I mean.

i Am not sure whether he know all the solutions. all his posts where only questions no replies or answers.
even if he knows all the answer , he should be interested in knowing different method of doing it.which usually people do less they are descendent's of Newton or Einstein .
 
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  • #5
Pi happens to be a transcendal no. meaning-you can't get its value by any algebraic method. If this a puzzle, you could get a nobel prize for the answer.
 
  • #6
Xalos said:
Pi happens to be a transcendal no. meaning-you can't get its value by any algebraic method. If this a puzzle, you could get a nobel prize for the answer.

That's not what transcendental means. Transcendental just means that it is not the root (zero) of any polynomial with integer coefficients. This has nothing to do with infinite sums such as this one, and there are many such sums (and products as well) that are quite well-known. The Wikipedia article on Pi and this MathWorld article list quite a lot of them.
 
  • #7
Xalos said:
If this a puzzle, you could get a nobel prize for the answer.
A Nobel Prize in what? Mathematics?
 
  • #8
Not to mention, The Nobel Prize is not award to Mathematicians unless the work has had a serious application in one of the awarded fields, such as the case of John Nash. I doubt this one would.

Anyone made any progress? It sounds like some of you already have a solution, just can't be bothered to post it >.<" Ill post mine tomorrow.

EDIT: CompuChip beat me to it !
 
  • #9
Hmm, can't be that hard if even I solved it in a little under 5 minutes. (Though I must admit, having the answer already helped me think of arctan a lot sooner :smile:)
 
  • #10
believe me it won't take more than a five min or even less than a min
 
  • #11
Aww don't say that! I wanted to sound really smart for having a solution! Ruin my fun :(
 
  • #12
thanks for all

from these http://mathworld.wolfram.com/PiFormulas.html


let[tex] S=1- \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - ... = \frac{\pi}{4}.[/tex]

thus [tex] \frac{1}{3} S= \frac{1}{3} - \frac{1}{9}+ \frac{1}{15} - \frac{1}{21} + ... = \frac{\pi}{12}.[/tex]

series = [tex] S + \frac{1}{3} S =\frac{\pi}{3} [/tex]
 
  • #13
Sorry for my answer! I am just a grade10 student and I just had read of the word in the library. I will remember not to reply about thing I don't know. Sorry Gib Z and thanks Moo of Doom!
 
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  • #14
Heh, interesting.
 
  • #15
that mine method too
 

1. What is a series in mathematics?

A series in mathematics is a sum of a sequence of terms. It can be represented as a1 + a2 + a3 + ... + an, where an denotes the terms of the sequence and n is the number of terms in the series.

2. What is the sum of a series?

The sum of a series is the result obtained when all the terms in the series are added together. It can be a finite or infinite value, depending on the terms in the series.

3. How do you prove the sum of a series?

To prove the sum of a series, you need to use mathematical techniques such as induction, telescoping, or the ratio test. These methods involve showing that the terms of the series approach a certain value or that the series converges to a specific value.

4. What is the value of pi/3?

The value of pi/3 is equal to approximately 1.0471975511965976. It is a rational number, which means it can be expressed as a ratio of two integers, but it is also an irrational number, which means it cannot be expressed as a finite decimal or fraction.

5. Why is proving the sum of this series equal to pi/3 significant?

Proving the sum of this series equal to pi/3 is significant because it shows a connection between two seemingly unrelated mathematical concepts - series and pi. It also highlights the beauty and complexity of mathematics and how seemingly simple equations can have profound implications.

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