Question

4^(x/y+y/x)=32 find X and y

Answer 1

By symmetry with the rest of the series, I have to suppose the first term is actually 1/(2·4) so the question is to find (we take a common 12 factor out of all terms)

12∑∞m=0∏mn=02n+12n+4

We can stick the inner part into Wolfram Alpha, just to see if it is reasonable to calculate, and we get that

∏mn=02n+12n+4=1π√Γ(m+32)Γ(m+3)

Once we know the answer, it is relatively easy to prove it, take out a common factor of 2 from each internal fraction, and multiply everything out:

∏mn=02(n+12)2(n+2)=∏mn=0n+12n+2=∏mn=0n+12(m+2)!

The product at the numerator is exactly Γ(m+32)Γ(12)=1π√Γ(m+32), by the multiplicative property of Γ.

So now we just have to find (again, take out the common 1π√ factor):

12π√∑∞m=0Γ(m+32)Γ(m+3)

If you plug ∑∞m=0Γ(m+32)Γ(m+3) back to Alpha, the answer is π−−√ , but in this case I haven’t been able to find out the actual operations to get to this result. One possible way is to use duplication formula to get  

Answer 2

Slander, defamation, or libel. There are legal distinctions among these terms,

☛ but they're all lies intended to hurt someone, and laws against uttering them publicly are considered to be one of the reasonable limitations on the constitutional freedom of expression.