Stirling's Formula
In permutations, we showed that the number of permutations of distinct objects is given by the factorial function How quickly does the factorial function grow as a function of This behavior is captured in the approximation known as Stirling's formula also known as Stirling's approximation.
Stirling's Formula
The factorial function is approximated by
Furthermore, for any positive integer , we have the bounds
What is the value of the following limit?
Central Binomial Coefficient
The theorem defined in binomial coefficient as for and it approaches asymptotically.
Given Stirling's formula, we have
Taking the ratio of the second approximation to the square of the first approximation,
Evaluate
Catalan Number
Catalan number occurs many times in various counting problems; it is defined to be . For more information, go to Catalan Numbers.
Find the limit
Note: For great honor, solve it without WolframAlpha.