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May 18, 2013

# Homework Help: Math(calculus)

Posted by Bash on Sunday, February 5, 2012 at 1:15am.

Evaluate lim n( In(n+1)- In n ) as n--> infinity

• Math(calculus) - drwls, Sunday, February 5, 2012 at 4:11am

Rewrite as
Lim ln(1 + 1/n)/(1/n)
(n -> infinity)

Then use the power series for ln(1+x) for small x:
ln(1 + 1/n) = 1/n -(1/2)(1/n)^2 +(1/3)(1/n)^3...

Divide that by 1/n and you get 1 plus a series of higher order terms that become zero in the limit as n-> infinity.

The limit is 1.

Check: Try n = 100
100*(ln101 -ln100) = 0.9950
n = 1000:
1000*(ln1001 - ln1000) = 0.9995

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