👉 Gregory's Math, also known as Gregory's Conjecture or the Gregory-Leibniz Inequality, is a fundamental result in real analysis that deals with the convergence of infinite series and integrals. It states that for any positive real number \(a\) and any integer \(n\), the series \(\sum_{k=0}^{n} \frac{(-1)^k a^{2k+1}}{(2k+1)!}\) converges and its sum is given by \(\frac{1}{2}a \ln\left(1 + \sqrt{1+a^2}\right)\). This inequality provides a powerful tool for estimating the sum of alternating series and has significant implications in various fields, including probability theory, numerical analysis, and physics. It essentially gives a bound on the error of approximating functions by their Taylor series expansions, making it a cornerstone in understanding the behavior of infinite sums and integrals.
