Estimating ln n and Proving its Limit as n Approaches Infinity

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SUMMARY

The discussion focuses on estimating the natural logarithm function, ln n, using the integral definition ln x = ∫ dx/x. Participants emphasize the importance of using upper and lower rectangular approximating sums to derive the best possible estimates for ln n, where n is a positive integer. The conclusion drawn is that as n approaches infinity, the limit of ln n also approaches infinity, confirming the unbounded growth of the natural logarithm function.

PREREQUISITES
  • Understanding of integral calculus, specifically the concept of definite integrals.
  • Familiarity with the properties of logarithmic functions.
  • Knowledge of upper and lower Riemann sums for estimating integrals.
  • Basic concepts of limits in calculus.
NEXT STEPS
  • Study the properties of logarithmic functions and their derivatives.
  • Learn about Riemann sums and their application in estimating integrals.
  • Explore the concept of limits in calculus, particularly with functions approaching infinity.
  • Investigate the relationship between logarithmic growth and polynomial growth rates.
USEFUL FOR

Students in calculus or mathematical analysis, educators teaching integral calculus, and anyone interested in the properties and applications of logarithmic functions.

kimberu
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Homework Statement



using the following definition: ln x = \int dx/x, give best possible upper and lower estimates of ln n, n a positive integer.

then, use this result to show that the limit of the function as n approaches infinity is infinity.


Homework Equations


--


The Attempt at a Solution


Unfortunately I have no idea what to attempt -- I don't know how to find estimates, and I'm not really sure what that means -- this is for a history of math class so I have no textbook context to look though either. Any explanations/tips would be very helpful, thanks!
 
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You probably mean to define

\ln{(x)} = \int_1^x\, \frac 1 t\, dt

Try looking at upper and lower rectangular approximating sums.
 

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