Answer
1n[∏ni=1(n+i)]1/n=[∏ni=11n(n+i)]1/n=[∏ni=1(1+in)]1/n
Taking log we get
1n∑ni=1ln(1+in)→∫10ln(1+x)dx,n→∞
Integrating by parts gives ∫10ln(1+x)dx=ln4−1.
Now the limit of the product is eln4−1.
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