How to Transform Slowly Converging Series into Rapidly Converging Series
Hengchun Mu ’20, Changchun, China
Majors: Quantitative Economics; Mathematics
The fundamental theorem of calculus states that differentiation and integration are inverse processes. From their definition, a derivative is a limit of difference and a integral is a limit of sum. With the application of theorems, finding derivatives could be as simple as finding integrals. However, finding differences is quite complicated even for a simple polynomial, x3, that is (x+1)3-x3. Is there any way to simplify the intensive calculation? The answer is yes.
Starting with proving if each term in a sum can be re-expressed as a difference of consecutive terms, then it can be summed, this presentation will mainly focus on 1/(x)_n case. Then it brings up the idea of the Stirling Numbers of the first kind and explain why the Stirling Numbers of the first kind actually matched with the numerators of the expansion of 1/(x)_(n+1). After verifying the accuracy of expansions with numerical examples, the last step is to test how good the approximations truly are. The approximations will be compared with the results generated from Euler-Maclaurin summation formula. The approximations are actually precise and the first eight decimals match the results.
The idea behind the application of the Stirling Numbers of the first kind is the transformation between difference and sum. After proving the equation that Stirling did not prove, this method will help fellow students or other scholars avoid intensive and time consuming calculations.