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Consider-the-series-I-n-1-e-x-lnx-n-dx-and-I-0-1-e-xdx-Which-of-the-following-is-true-a-0-I-n-e-2-n-2-b-1-I-n-e-2-n-1-c-I-n-is-negative-




Question Number 113865 by Ar Brandon last updated on 15/Sep/20
Consider the series I_n =∫_1 ^e x(lnx)^n dx and I_0 =∫_1 ^e xdx  Which of the following is true ?  a\ 0≤I_n ≤(e^2 /(n+2))    b\1≤I_n ≤(e^2 /(n+1))  c\I_n  is negative
$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{series}\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{1}} ^{\mathrm{e}} \mathrm{x}\left(\mathrm{lnx}\right)^{\mathrm{n}} \mathrm{dx}\:\mathrm{and}\:\mathrm{I}_{\mathrm{0}} =\int_{\mathrm{1}} ^{\mathrm{e}} \mathrm{xdx} \\ $$$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{true}\:? \\ $$$$\mathrm{a}\backslash\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\frac{\mathrm{e}^{\mathrm{2}} }{\mathrm{n}+\mathrm{2}}\:\:\:\:\mathrm{b}\backslash\mathrm{1}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\frac{\mathrm{e}^{\mathrm{2}} }{\mathrm{n}+\mathrm{1}}\:\:\mathrm{c}\backslash\mathrm{I}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{negative} \\ $$

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