Standard and Less Standard Asymptotic Methods

Explore advanced mathematical techniques for evaluating infinite sums and analyzing asymptotic behavior of sequences through this comprehensive lecture series. Master both standard methods like the Euler-Maclaurin formula and its variants, alongside less conventional approaches for tackling complex numerical problems. Learn to evaluate slowly convergent infinite sums to hundreds of decimal places, determine precise asymptotic laws for sequences when given only initial values, and compute highly oscillatory series with large parameters to high accuracy. Work through practical examples including evaluating binomial coefficient sums, expanding infinite products as power series to identify asymptotic constants, and computing convergent series with complex arguments. Develop skills in recognizing exact values of computed constants and applying various asymptotic methods to real mathematical problems encountered across different branches of mathematics. The series provides extensive examples and applications, demonstrating how these techniques solve challenging numerical evaluation problems that arise in advanced mathematical research.