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what is the value of integration limit n-> infinity [n!/n to the power n]to the power 1/n

what is the value of integration limit n-> infinity [n!/n to the power n]to the power 1/n

Grade:12

3 Answers

Akash Kumar Dutta
98 Points
8 years ago

limit n-->inf.    [1 + (n!-n^n)/n^n]^1/n

= e^ limit n-->inf.    {(n!-n^n)/n^n}.1/n         

                             (applying formula lim x-->a  [1 + f(x)]^g(x) =    e ^ {lim x-->a    f(x).g(x)}  )

=e ^ lim n-->inf.       (n-1)!/(n)^n-1   -   1/n

=e ^ lim n---> inf.     (1-1/n)(1-2/n)(1-3/n)......1/n   -    1/n

=e ^ (1-o)(1-0).....(1-0).0  -   0

=e ^ 0-0

=1

hence, integation 1= x + C (ANS)

hope i helped you.

gauhar singh
7 Points
8 years ago

e-1/2

Akash Kumar Dutta
98 Points
8 years ago

the ans is
-e^-1

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