Friday, 15 March 2019

limits - What is wrong with the following "proof" that e=1?




Let's analyze this expression



lim



It's the definition of e which, as we know is not equal to 1. So what is wrong with the following "logic":



As \lim_\limits{n\rightarrow\infty}(a_{n}b_{n}) = \lim_\limits{n\rightarrow\infty}(a_{n})\times\lim_\limits{n\rightarrow\infty}(b_{n}) and \lim_\limits{n\rightarrow\infty}(1+\frac{1}{n}) = 1, we can say that \lim_\limits{n\rightarrow\infty} (1+\frac{1}{n})^n=1^n, which is equal to 1.



I know something's wrong there, but the question is - what?



Answer



What you actually proved is that
\lim_{k\to\infty} \left(\lim_{n\to\infty} 1+ \frac1n\right)^k = 1
Wich is correct, but the LHS is not equal to e.
The problem is most apparent when you end up with a 1^n (supposed to be a \lim_{k\to\infty} 1^k) and got rid of the limit expression \lim_{n\to\infty}.


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