Monday, 30 June 2014

sequences and series - Finding limit of a product.



Prove:limn1n[ni=1(n+i)]1n=4e


I tried using Squeeze Theorem but can't go beyond $1

Answer



1n[ni=1(n+i)]1/n=[ni=11n(n+i)]1/n=[ni=1(1+in)]1/n

Taking log we get
1nni=1ln(1+in)10ln(1+x)dx,n
Integrating by parts gives 10ln(1+x)dx=ln41.
Now the limit of the product is eln41.


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