TWISTING EQUATIONS FROM DISCRETE SUMS TO CONTINUOUSTRUTHS

ABSTRACT. I explore the sequence 𝑄𝑛 = 1𝑛 ∑𝑛1𝑘=1√(𝑘/𝑛)2+(1−𝑘/𝑛)2 , transforming a discretesum into a continuous integral. Using the Euler-Maclaurin formula, I uncover its limitas √2 ln(1+√2), with precise correction terms and error bounds. Numerical experimentsconfirm my results with accuracy, and I extend the framework to a broader class of sums,including multi-dimensional cases<br>