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Prove that`int_0^oo[n e^(-x)]dx=1n((n^n)/(n !)),w h e r en`is a natural number greater than 1 and [.] denotes the greatest integerfunction.. |
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Answer» `AAxe[0,oo),n e^(-x)epsilon(0,n]` If `0ltn e^(-x)1,xepsilon(In n,oo)`. If `1le n e^(-x)lt 2,xepsilon(In n//2, In n]` If `2le n e^(-x)lt3,xepsilon(In n//3, In n//2]` If `n-1le n e^(-x)ltn,xepsilon(0,In n/(n-1)]` `:. int_(0)^(oo) [n e^(-x)]dx=int_(0)^(In n/(n-1))(n-1)dx+int_(In n/(n-1))^(In n/(n-2))(n-2)dx` `+............+int_(In n/2)^(In n) 1dx+int_(In n)^(oo)0 dx` `=(n-1)(In n/(n-1))+(n-2)[In (n/(n-2))` `-In(n/(n-1))]+..............+1[In n-In-n/2]` `=In((n^(n))/(n!))`. |
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