Question

Given: An = [6 n/(-4 n + 9)] For both of the following, decide whether the given sequence or series is convergent or divergent. (a) The sequence {An}. (b) The series sum n=1 to infinity (An).

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 1212 factor out of all terms)

12∑∞m=0∏mn=02n+12n+412∑m=0∞∏n=0m2n+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)∏n=0m2n+12n+4=1πΓ(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)!∏n=0m2(n+12)2(n+2)=∏n=0mn+12n+2=∏n=0mn+12(m+2)!

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

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

12π√∑∞m=0Γ(m+32)Γ(m+3)12π∑m=0∞Γ(m+32)Γ(m+3)

If you plug ∑∞m=0Γ(m


+32)Γ(m+3)∑m=0∞Γ(m+32)Γ(m+3) back to Alpha, the answer is π−−√π , but in this case I haven’t been able to find out the a


ctual operations to get to this result. One possible way is to use duplication formula to get Γ(m+32)=π√Γ(2m+2)22n+1Γ(m+1)Γ(m+32)=πΓ(2m+2)22n+1Γ(m+1) so the fraction becomes (as Γ(n+1)=n!Γ(n+1)=n!):

π−−√∑∞m=0(2m+1)!22m+1m!(m+2)!π∑m=0∞(2m+1)!22m+1m!(m+2)!

In any case the raw result is