limit

lim { (1/n)((2n+1)(2n+2)2n+3)............(2n+n)^1/n))}
n-->âˆž

2 Answers

1357

Manish Shankar ·2009-10-02 22:44:02

Hint: (1/n)((2n+1)(2n+2)2n+3)............(2n+n)^1/n)
=[(2+1/n)(2+2/n)(2+3/n)...........(2+n/n)]^1/n

106

Asish Mahapatra ·2009-10-02 22:50:57

now take log on both sides

so that
logy = Î£(1/n)log(2+r/n)
= ₀âˆ«¹log(2+x)dx

now use parts and simplify

logy = [(x+2)log(2+x) - x]₀²

logy = [4log4-2 - 2log2]

logy = 6log2-2
= log(2⁶/e²)
So, y = 2⁶/e²

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