Rigorous Proof lim(1 + x/n)^n Equals e^x for All Real x. скачать в хорошем качестве

Rigorous Proof lim(1 + x/n)^n Equals e^x for All Real x.

#RealAnalysis #LimitProof #ExponentialFunction #RigorousMath #MathProof #Calculus #AdvancedCalculus #MathEducation #PureMath #MathematicalLimits Prerequisites: The best is to watch the full playlist:    • Proof By Induction: All the horses are of ...   The minimal prerequisites are the limit e=(1+1/n)^n can be found here:    • Euler's number e=lim(1+1/n)^n, Base of Nat...   and the properties of real exponent, where we prove lim a_n^b_n=A^B , which is the previous video in this playlist:    • Rigorous Foundations of Real Exponents and...   In this video, we rigorously prove that the limit of (1 + x/n)^n as n approaches infinity equals exp(x) for every real number x. This fundamental identity is central to calculus, real analysis, and the definition of the exponential function. Unlike informal proofs restricted to positive x, we provide a general and complete argument that holds for all real values of x. What you'll learn: Why the limit works for all real x, not just positive values How to connect the expression (1 + x/n)^n to the power series of exp(x) A step-by-step, rigorous analysis of convergence How error bounds and inequalities guarantee this identity. This proof is suitable for students of advanced calculus, real analysis, or anyone seeking a mathematically sound understanding of one of the most important limits in mathematics. No shortcuts. No hand-waving. Full rigor.