Number e - Calculus 1

The base shrinks toward 1. The exponent runs off to infinity. One of those effects should win — and neither does. What survives is 2.718281828..., and this video builds it from nothing. Nothing tells you e exists except the monotone convergence theorem. This is that theorem's first real payment: we prove the sequence increases, prove it never passes 3, and only then are we allowed to say the limit is there and give it a name. There is no closed form hiding behind e. The limit IS the definition. Then the part nobody warns you about: the definition of e and a good way to compute e are not the same object. The limit crawls. The factorial series falls off a cliff. Ten terms of one beats a million terms of the other. Ends with four worked limits, including all three from the lesson's Check Yourself list. CHAPTERS 0:00 Intro 0:05 The standoff: base to 1, exponent to infinity 0:38 The theorem: increasing, bounded, therefore e 1:20 Why it never passes 3 2:01 Why it never drops below 2 2:29 The sandwich: two sequences closing on e 3:10 Two routes, wildly different speeds 3:51 The generalisation you will actually use 4:29 Worked example: ((n+3)/(n+1))^n 5:17 Three exercises, solved 6:16 Synthesis WHAT YOU NEED FIRST Monotone sequences (increasing + bounded = convergent), and the squeeze theorem. Basic interval and limit notation.