Another fundamental infinite product & summation

[Loren Booda]

Can you find the values of

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

and

[oo]
[sum] sin(((-1ⁿ⁺¹)(2n+1)!)^1/(2n+1))
n=0

?

 

[bogdan]

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"...

 

[Loren Booda]

bogdan-

"Fundamental" because I have so dam much fun thinking them up. You confirmed that two of my first four series may indeed be fundamental, and that they (L & B), although non-trivially related to each other at first glance, give similar answers. These here are my last, and I am graciously interested whether they, too, establish a pattern. Again, you did me a great favor by establishing numerically the first connection. I may be able using the Taylor series to make such a connection, but my computing ability is primitive (an old TI scientific calculator) for such an app. Thanks for your interest and diligence.

 

[bogdan]

Well...it's not that simple now...because you have (2n+1)!, which for n>6 is too big for long variable (only 32 bits) so I'll have to implement some function to work with huge numbers (thousands of '1','2',...(I don't know the word )...

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...

 

[Loren Booda]

bogdan-

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]?)

 

[bogdan]

Nice theorem...
If I may...how did this idea come to life ?
How did you think about it ?

 

[Loren Booda]

bogdan-

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.

 

[bogdan]

Wow...America is truly the country of all posibilities...
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...

 

[Loren Booda]

Olympiads sound more challenging than I could withstand. I got a perfect score (800) on my high school math SAT's, but that just shows an aptitude in test-taking. I prefer lying back with an equation in my head and ruminate. "Freedom" does have its advantages. I am fortunate enough to work mostly for social concerns - my Alzheimer's group, an 800-mental health line, and a local park. I feel like I am helping the world more directly with these rewarding, volunteer tasks.

 

[bogdan]

Olympiads have their advantages...for example not going to school before them (aprox 1 month)-> plenty of time to "relax" and think about other problems... (mathematical problems)