limn→∞n√n!=limn→∞n√1∗n√2⋯⋅n√n=1⋅1⋅…⋅1=1
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=limn→∞nn=limn→∞1+1+…+1n=limn→∞1n+1n+…+1n=0+0+…+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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