It is e to the iπ week over here at @polymathematic HQ. All week long I will be publishing videos getting at an intuition for what on earth it might mean to raise e to an imaginary power, how the π fits into all that, and why we should expect that to equal –1.
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To begin, I'm going to show how we can evaluate the limit as n grows toward infinity of (1 + 1/n)^n. This is the limit definition of the first power of e, and it gives us a way to define exponentiation without solely relying on repeated multiplication. For that matter, it also gives us a way to compute continuous compound interest!
Proving this limit requires the use of L'Hopital's Rule, also known as Bernoulli's rule, which is fitting, since Jacob Bernoulli was one of the earliest mathematicians, alongside Euler, to prove this result about e. We can use the natural logarithm to manipulate the original expression into an indeterminate form, and apply L'Hopital from there.
#realanalysis #calculus #lhopitalsrule
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