r/askmath • u/Equivalent_Dingo_683 • 9d ago
Discrete Math Simplify sum
Friend gave this "easy" sum for Olympiad practice, can't solve it even though I already tried relating it to e's Taylor series done harmonic stuff etc, any help?
•
u/Bounded_sequencE 9d ago
What exactly is the assignment? Also, please provide a link to the official olympiad paper, to ensure this is is a legit question.
•
u/Equivalent_Dingo_683 9d ago
friend made it up and claims to have a closed solution but isn't giving it
•
u/Bounded_sequencE 9d ago
Chances are high you have been trolled successfully^^
You also did not answer the question what exactly the assignment is.
•
u/AppropriateCar2261 9d ago
You probably have a typo. I gave this sum to Wolfram Mathematica, and even it couldn't find a simpler expression.
•
u/Bounded_sequencE 9d ago edited 9d ago
It's fairly easy to find an upper estimate to show the series tends to zero as "n -> oo". That's about the only "nice" pattern I've been able to find so far.
•
•
u/veryjewygranola 9d ago
If you define your sum up to n to be S(n), and look at the ratios of S(n)/S(n-1), you see it approaches 1/2 rather quickly, so we can approximate the asymptotic behavior of S(n) as 2^(-n+1) (we do n+1 since S(2) = 1/2).
A closed form or exact expression for this sum is unlikely.
•
u/Mission_Rice3045 9d ago
Hmm, what happens if you take the log of the sum terms? This would turn the factorial into a sum
•
u/Bounded_sequencE 9d ago
Logarithms are not linear -- applying "ln(..)" to the entire sum does not carry over to the argument.
•
u/Mission_Rice3045 9d ago
Yes, but (...) = e^{ln(...)}. Like I said this allow you to write the factorial as a sum. Don't know if this will meaningfully simplify this summation but it is something commonly done.
•
•
•
u/uTRexAap 9d ago
havent taken any calc so im terrible at this but i got it down to this