- #1

- 3,106

- 4

Can you find the values of

[oo]

n=0

and

[oo]

[sum] sin(((-1

n=0

?

[oo]

_{[pi]}cos(((-1^{n})(2n)!)^{1/(2n)})n=0

and

[oo]

[sum] sin(((-1

^{n+1})(2n+1)!)^{1/(2n+1)})n=0

?

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- Thread starter Loren Booda
- Start date

- #1

- 3,106

- 4

[oo]

n=0

and

[oo]

[sum] sin(((-1

n=0

?

Last edited:

- #2

- 191

- 0

From where do you take these things ?

Can't the second sum be taken term by term and "Taylored" around 0

(sinx =x-x^3/3!...) and then rearrange these terms (because if we presume that the sum is convergent we can do that) and obtain something...but if this is a "fundamental" fact then it will be better if you can tell us more about these "sums"...

Can't the second sum be taken term by term and "Taylored" around 0

(sinx =x-x^3/3!...) and then rearrange these terms (because if we presume that the sum is convergent we can do that) and obtain something...but if this is a "fundamental" fact then it will be better if you can tell us more about these "sums"...

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- #3

- 3,106

- 4

"Fundamental" because I have so dam much fun thinking them up.

- #4

- 191

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Have you tried to prove that the sin sum is convergent ? (Cauchy or D'Alembert...) because I'll have a headache if I try...

Isn't that sin sum the series development (I don't have the english expression for this) of a function ? Or something like the Fourier "transformate" (bad english...) ?

Anyway...if you'll wait 2 weeks I think I'll give you the answers because I'll meet some friends who are some of the best in my country (at their age) in math analysis so...maybe they'll know...

- #5

- 3,106

- 4

You have been very tolerant to consider my musings. Please accept my thanks for your genuine interest. If only you would introduce "L" and "B" to your friends. The other series in my posts, like most mathematical attempts, seem effete. I wish much beauty for you to find in mathematics. (Have you seen the "Booda Theorem," on my website, [through the www button, below]?)

- #6

- 191

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Nice theorem...

If I may...how did this idea come to life ?

How did you think about it ?

If I may...how did this idea come to life ?

How did you think about it ?

- #7

- 3,106

- 4

In my pre-calc class in 11th (junior) grade in high school, a smart jock (athelete) Dewey Allen found the numerical pattern while my teacher worked out solutions to polynomials on the board. My teacher then challenged the class (particularly Booda) to come up with a general theorem. Not yet 17, I crunched variables, and solved it once I realized [del]f(x) was exactly divisible by [del]x. I then entered it into our county science fair, and received second place in mathematics. I'm now 44, and have never again completed another math proof. I believe that a similar proof is doable at least for quartics, but I'll leave that up to you. Solving the general case for polynomial of arbitrary rank n should get one some notoriety.

- #8

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We don't have here in Romania such contests...where you can show you work (new theorems...things like those)...

Instead we have stupid contests (olympiads) where you have to solve 4 problems in 3 hours... My brain takes fire...

- #9

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- #10

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