I've tried and searched for a long time, and I haven't been able to prove or find a proof that the following sequence converges (without using another definition of the exponential function):(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\forall x \in \mathbb{R}[/tex]. Prove that:

[tex]\lim_{n \rightarrow \infty} (1+ x/n)^n[/tex] exists.

I can prove that it is monotonic and using binomial theorem I can show that it is bounded for x=1. However if I try to use the same approach for general x, I get the power series for e^x and can only say it is bounded if I can prove the power series converges, which I don't know how to do. But even if I did is there any way to prove this limit exists without proving the power series converges (the other definition of e^x)?

Another related question I can't figure out is, how can I prove that

[tex]\lim_{n \rightarrow \infty} (1+ x/n + o(x/n))^n = \lim_{n \rightarrow \infty}(1 + x/n)^n [/tex]

where [tex] o(x/n) [/tex] is any function that goes to 0 more quickly as [tex](x/n) \rightarrow 0 [/tex] than [tex](x/n) [/tex]?

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# How to prove definition of exponential function as limit of powers converges

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