MHB Fundamental theorem and limit proofs

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The discussion focuses on proving that the limit of ((2^n * n!)/n^n) approaches zero as n approaches infinity. Participants suggest using Stirling's approximation to simplify the factorial term, which helps in analyzing the logarithm of the limit. By applying the continuity of the logarithm function, the logarithm can be pulled inside the limit for easier manipulation. The approximation of log(n!) using Stirling's formula allows for simplification, leading to a conclusion about the limit. Ultimately, the discussion emphasizes the importance of careful logarithmic manipulation and approximation techniques in limit proofs.
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Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero.

The hint is to use Stirling's approximation. What is this?
 
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Take the logarithm of the limit, and pull the logarithm inside the limit by using the continuity of the logarithm function. What are you left with? Can you approximate $\log(n!)$ using Stirling's approximation? Simplify, and then you should be able to conclude (don't forget to undo the logarithm you applied to the limit at the start).
 

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