Fundamental theorem and limit proofs

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SUMMARY

The limit as n approaches infinity of ((2^n * n!)/n^n) equals zero, as established through the application of Stirling's approximation. By taking the logarithm of the limit and utilizing the continuity of the logarithm function, one can simplify the expression. The approximation of $\log(n!)$ using Stirling's formula is crucial in this proof. Ultimately, the logarithmic transformation allows for a straightforward conclusion that the original limit approaches zero.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with Stirling's approximation
  • Knowledge of logarithmic functions and their properties
  • Basic combinatorial concepts related to factorials
NEXT STEPS
  • Study Stirling's approximation in detail
  • Learn about logarithmic transformations in limit proofs
  • Explore advanced limit techniques in calculus
  • Practice problems involving factorial growth rates
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Mathematics students, educators, and anyone interested in advanced calculus and limit proofs will benefit from this discussion.

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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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