Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 138159 by BHOOPENDRA last updated on 10/Apr/21

Answered by Dwaipayan Shikari last updated on 10/Apr/21

∫_0 ^1 x^(−x) dx=ℵ  =∫_0 ^1 e^(−xlog(x)) dx=Σ_(n=0) ^∞ (((−1)^n )/(n!))∫_0 ^1 x^n log^n (x)dx=ℵ  ∫_0 ^1 x^n dx=(1/(n+1))⇒∫_0 ^1 x^n log^m (x)dx=(∂^m /∂n^m )((1/(n+1)))=((m!(−1)^m )/((n+1)^(m+1) ))  But here m=n   so  ℵ=Σ_(n=0) ^∞ (((−1)^n )/(n!)).(((−1)^n n!)/((n+1)^(n+1) ))=Σ_(n=1) ^∞ (1/n^n )

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{−{x}} {dx}=\aleph \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{xlog}\left({x}\right)} {dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {log}^{{n}} \left({x}\right){dx}=\aleph \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {dx}=\frac{\mathrm{1}}{{n}+\mathrm{1}}\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {log}^{{m}} \left({x}\right){dx}=\frac{\partial^{{m}} }{\partial{n}^{{m}} }\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)=\frac{{m}!\left(−\mathrm{1}\right)^{{m}} }{\left({n}+\mathrm{1}\right)^{{m}+\mathrm{1}} } \\ $$$${But}\:{here}\:{m}={n} \\ $$$$\:{so}\:\:\aleph=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}.\frac{\left(−\mathrm{1}\right)^{{n}} {n}!}{\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} }=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{n}} } \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com