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Let a(n) = (1+1/n)^(n) . Then for each n...

Let `a_(n) = (1+1/n)^(n) .` Then for each `n in N`

A

`a_(n)ge 2`

B

`a_(n)lt 3`

C

`a_(n)lt 4`

D

`a_(n)lt 2`

Text Solution

Verified by Experts

The correct Answer is:
a, b, c
`because a_(n) = (1+1/n)^(n) = ""^(n) C_(0) + ^(n) C_(1) cdot (1/n) + sum_(r=2)^(n) ""^(n)C_(r)(1/n)^(2)`
`=2 + sum_(r=2)^(n)""^(n)C_(r) (1/n)^(2)`
`therefore a_(n) ge 2 `for all `ninN`
`lim_(nrarrinfty)(1+1/n)^(n)=e=2.7182...`
`therefore a_(n) lt e `
Finally, `2le a_(n) lte`
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