What Is the Limit of n! / n^n as n Approaches Infinity?

TL;DR

The limit of n! / n^n as n approaches infinity equals 1/e. By using logarithmic expansion and the concept of the area under the curve, the evaluation reveals that the final result is derived through integration, leading to this surprising outcome.

Transcript

let's do somewhere for fun here we have the limit as n goes to infinity M factorial over N to the nth power and then raised to the 1 over N power as you can see this looks pretty crazy right and you can also see from the screenshot right here I got this question from brilliant at work back in 2015 and right now I'm really happy to show you guys how... Read More

Key Insights

• 🔨 Logarithmic expansion can be a useful tool in solving complex mathematical problems.
• 🥺 The concept of area under a curve can help simplify equations and lead to the evaluation of integrals.
• 🤪 The limit as n goes to infinity can have unexpected results, such as 1/e.

Questions & Answers

Q: What is the equation being discussed in the video?

The equation is the limit as n goes to infinity of n factorial over N to the nth power, raised to the 1 over N power.

Q: How does the video simplify the equation using logarithmic expansion?

By introducing the variable L and taking the natural log of both sides, the equation is transformed into a product of natural logs, simplifying the problem.

Q: What concept is used to further simplify the equation into an integral?

The concept of area under a curve is used, where the equation is transformed into the integral from 0 to 1 of Ln X dx.

Q: What is the final solution to the limit as n goes to infinity?

The final solution is 1/e, where e is the base of natural logarithms.

Summary & Key Takeaways

• The video explores solving the limit as n goes to infinity in a complex equation using logarithmic expansion.

• The concept of area under a curve is introduced to simplify the equation.

• Through the process of integration by parts, the antiderivative of Ln X is found, leading to the solution of 1/e.

• The video also promotes the website Brilliant at Work, which offers challenging math problems and step-by-step courses.