Hello. This is a nice video, thanks for sharing. I have however a comment about the proof that the two e definitions are the same (which of course they are!). At 15:30, one cannot just say that the limit of the sum is the sum of the limits, because the sum contains a *variable* number of terms, which itself depends on n. Let me make a different example. Suppose we have to compute the limit, for n → +∞, of the sum (1/n + 1/n + 1/n + ··· 1/n), which contains n identical terms 1/n. Of course the sum of the n terms is n·1/n = 1 for every n, and so the limit is 1. But if I were to take the sum of the limits, I would get 0 + 0 + ··· = 0, which is wrong. Now, in this case, the trick works, and of course the equality lim (1+1/n)ⁿ = ∑1/n! is correct, but the proof does not justify the exchange of the limit with the summation. A possible way of doing this is resorting to Tannery's theorem (which is the "series" version of the dominated convergence theorem for exchanging limits with integrals).
@@uggupuggu The function I'm referring to is the exponential function. It can be defined as a power series or a limit, and does not involve the base of the natural logarithm (the number e).
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Such an awesome video!
❤️❤️❤️
Hello. This is a nice video, thanks for sharing. I have however a comment about the proof that the two e definitions are the same (which of course they are!). At 15:30, one cannot just say that the limit of the sum is the sum of the limits, because the sum contains a *variable* number of terms, which itself depends on n.
Let me make a different example. Suppose we have to compute the limit, for n → +∞, of the sum (1/n + 1/n + 1/n + ··· 1/n), which contains n identical terms 1/n. Of course the sum of the n terms is n·1/n = 1 for every n, and so the limit is 1. But if I were to take the sum of the limits, I would get 0 + 0 + ··· = 0, which is wrong.
Now, in this case, the trick works, and of course the equality lim (1+1/n)ⁿ = ∑1/n! is correct, but the proof does not justify the exchange of the limit with the summation. A possible way of doing this is resorting to Tannery's theorem (which is the "series" version of the dominated convergence theorem for exchanging limits with integrals).
Thanks Sir❤️❤️❤️
e is actually a function, e(x), with the numerical value of the number e computed at x=1.
e is a number
the function you are referring to is e^x
@@uggupuggu The function I'm referring to is the exponential function. It can be defined as a power series or a limit, and does not involve the base of the natural logarithm (the number e).
Excellent 🔥 we need more sequence and series with infinity concepts, also integration
❤️❤️❤️🙏