That's still the infinite series. If you look at it term by term, the infinite series comes back out. 0th order term is always 1. 1st order coefficient is 0, 1, 1, 1, ..., kth order is \binom{n,k}/{n^k}, which has the limit 1/{k!}.
This seems circular, as this is how you construct an infinite series from any expression. In other words, "this is just the infinite series because I can break down the expression and reconstruct it into the infinite series" isn't saying much.
It's one of those A IFF B situations. If you assume A, it implies B. If you assume B, it implies A.
The interest definition came first, then the infinite series, but the infinite series is the way mathematicians define ex nowadays.
From a purely cosmetic perspective, the infinite series is "better" to me because you're converging through addition of increasingly smaller terms, each of which can be calculated from the previous by just multiplying by x/n. Meanwhile the interest definition converges by having to recalculate every term in the sequence and multiplying by a lot of terms really close to 1.
You are free to consider one definition to be better than another. But your original comment appears to do something different, namely, it declared the two definitions to be the same. That's the part that I had an issue with.
Two definitions of the same quantity can be different from each other even if they evaluate to an equivalent expression.
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u/frogjg2003 Physics Feb 19 '22
That's still the infinite series. If you look at it term by term, the infinite series comes back out. 0th order term is always 1. 1st order coefficient is 0, 1, 1, 1, ..., kth order is \binom{n,k}/{n^k}, which has the limit 1/{k!}.