MHB A generalization of the limit definining \$e\$.

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The discussion explores the generalization of the limit defining the exponential function, specifically focusing on the expression $\lim_{n_1\to \infty , n_2 \to \infty , \ldots , n_k \to \infty } (1+\prod_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$, which simplifies to $\exp(\prod_i x_i)$. The author then shifts to consider a more complex case involving addition: $\lim (1+\sum_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$, expressing uncertainty about how to approach this limit. The discussion invites insights on solving this more challenging limit involving sums rather than products. Overall, the thread centers on the exploration of limits related to the exponential function and seeks further mathematical guidance.
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I was thinking of generalizing the limit of $\lim_{n\to \infty} (1+x/n)^n=\exp(x)$. What do we know of $$\lim_{n_1\to \infty , n_2 \to \infty , \ldots , n_k \to \infty } (1+\prod_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$$?
 
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Well, now I think it's trivial if we define $m=\prod_i n_i$, then the limit should be: $\exp(\prod_i x_i)$, nothing new here.

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How about instead of multiplication we have addition, i.e.:
$\lim (1+\sum_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$, which seems tougher to find.
How would you go about solving this limit?

Thanks!
 
There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...

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