Saturday, 3 September 2016

calculus - What is wrong with this fake proof that limlimitsnrightarrowinftysqrt[n]n!=1?



lim
I already know that this is incorrect but I am wondering why. It probably has something to do with the fact that multiplication in n! is done infinite number of times.


Answer



Start by figuring out a simpler example:
1 = \lim_{n\to\infty} \frac n n = \lim_{n\to\infty} \frac {1+1+\ldots+1} n = \lim_{n\to\infty} \frac 1 n + \frac 1 n + \ldots + \frac 1 n = 0 + 0 + \ldots + 0 = 0



Indeed, you cannot exchange sum (or product) and limit if the amount of terms in the sum or product depend on the limiting variable.


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