“This Series Can Equal ANY Number (Riemann Rearrangement Theorem)” скачать в хорошем качестве

“This Series Can Equal ANY Number (Riemann Rearrangement Theorem)”

This infinite series converges — but you can rearrange it to equal ANY real number. In this video, we analyze the classic alternating harmonic series ∞ ∑ (-1)^n/n 𝑛=1 and show why conditional convergence completely breaks the intuition most students have about infinite sums. Using only the Riemann Rearrangement Theorem, we uncover: why convergence alone is not enough, how positive and negative terms secretly compete, and how rearranging terms lets you force the series to converge to any value you want. This idea shows up in: real analysis Fourier series mathematical physics and why rigor actually matters in calculus and beyond. 🧩 Who this video is for Strong Calculus II / III students Real Analysis beginners Physics & engineering students using series in practice Anyone who has ever been told “don’t worry, it converges” 📚 Resources 📘 My book (for strong undergrads): Advanced Integration Techniques 👉 https://www.stem1online.com/category/... 🎓 1-on-1 Calculus, ODEs & Physics tutoring: (Links in description / pinned comment) 🔔 Why this matters If you don’t understand this example, Fourier series, integrals, and physics expansions will eventually betray you. This video fixes that — cleanly and visually. 00:00 – Why this problem is harder than it looks 01:02 – The symmetry everyone misses 02:10 – Why integrating the electric field fails 03:28 – Switch to electric potential (the key idea) 05:10 – Computing the potential cleanly 06:45 – Differentiating to get the electric field 08:20 – Physical interpretation of the result 09:30 – How this generalizes to disks and planes 10:45 – What to remember for exams #RealAnalysis #InfiniteSeries #AdvancedMath #STEMEducation #Mathematics