Efetue o cálculo a seguir:
[tex] \Large{ \sf{\displaystyle\int^{\pi}_{0} {e_{1}} ^{{ e_{2} }^{{ \dots} }}}^{{e_{n - 1}} ^{{ e_{n} }^{ix} } } dx}[/tex]
PS: Desde já agradeço.
[tex] \Large{ \sf{\displaystyle\int^{\pi}_{0} {e_{1}} ^{{ e_{2} }^{{ \dots} }}}^{{e_{n - 1}} ^{{ e_{n} }^{ix} } } dx}[/tex]
PS: Desde já agradeço.
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Siga o passo abaixo:
[tex]\large{ \sf{\displaystyle\int^{\pi}_{0} {e_{1}} ^{{ e_{2} }^{{ \dots} }}}^{{e_{n - 1}} ^{{ e_{n} }^{ix} } } dx}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\int^{\pi}_{0}\sf{{e_1}^{{a_1e_2}^{{.}^{{.}^{{.}^{{e_{{n-2}}^{{e_{n-1}^{ix}}}}}}}}}}dx}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}}\displaystyle\int^{\pi}_{0}\sf{{e_1}^{{a_2e_2}^{{.}^{{.}^{{.}^{{e_{{n-3}}^{{e_{n-2}^{ix}}}}}}}}}}dx}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}}\displaystyle\sum^{\infty}_{\sf{a_3=0}}\sf{\cfrac{{a^{a_3}_2}}{a_3 !}} \displaystyle\int^{\pi}_{0}\sf{{e_1}^{{a_2e_2}^{{.}^{{.}^{{.}^{{e_{{n-3}}^{{e_{n-2}^{ix}}}}}}}}}}dx}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}}\displaystyle\sum^{\infty}_{\sf{a_3=0}}\sf{\cfrac{{a^{a_3}_2}}{a_3 !}} \ldots{\displaystyle\sum^{\infty}_{\sf{a_{n-2}=0}}\sf{\cfrac{a^{{a_n-2}}_{n-3}}{a_{n-2} !}}\displaystyle\sum^{\infty}_{\sf{a_{-1}=0}}\sf{\cfrac{{{{a^{a_{n-1}}_{n-2}}}}}{a_{n-1}!}}}\displaystyle\int^{\pi}_{0}\sf{e^{ia_{n-1}x}dx}}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}}\displaystyle\sum^{\infty}_{\sf{a_3=0}}\sf{\cfrac{{a^{a_3}_2}}{a_3 !}} \ldots{\displaystyle\sum^{\infty}_{\sf{a_{n-2}=0}}\sf{\cfrac{a^{{a_n-2}}_{n-3}}{a_{n-2} !}}\displaystyle\sum^{\infty}_{\sf{a_{-1}=0}}\sf{\cfrac{{{{a^{a_{n-1}}_{n-2}}}}}{a_{n-1}!}}}\displaystyle\int^{\pi}_{0}\sf{\cos\left( a_{n-1}x\right)dx}}[/tex]
[tex]\large{\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}}\displaystyle\sum^{\infty}_{\sf{a_3=0}}\sf{\cfrac{{a^{a_3}_2}}{a_3 !}} \ldots{\displaystyle\sum^{\infty}_{\sf{a_{n-3}=0}}\sf{\cfrac{a^{{a_n-3}}_{n-4}}{a_{n-3} !}}\displaystyle\sum^{\infty}_{\sf{a_{n-2}=0}}\sf{\cfrac{{{{a^{a_{n-2}}_{n-3}}}}}{a_{n-2}!}}}\pi}[/tex]
[tex]\large\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1}!\:\ldots\:\displaystyle\sum^{\infty}_{\sf{a_{n-k}=0}}\sf{\cfrac{a^{n-k}_{n-k-1}}{a_{n-k}!}}\left(\sf{{e^{e^{.^{.^{.^{e^{e_{n-2}}_{n-3}}}}}_2}_1}} \right)^{a_{n-k}}\pi}[/tex]
[tex]\large\displaystyle\sum^{\infty}_{\sf{a_1=0}}\sf{\cfrac{1}{a_1 !}}\displaystyle\sum^{\infty}_{\sf{a_2=0}}\sf{\cfrac{{a^{a_2}_1}}{a_2 !}\left( \sf{{e^{e^{.^{.^{.^{e^{e_{n-4}}_{n-5}}}}}_2}_1}} \right)\pi}[/tex]
[tex]\large\displaystyle\sum^{\infty}_{\sf{a_1=0}\sf{\cfrac{\left( \sf{{e^{e^{.^{.^{.^{e^{e_{n-3}}_{n-4}}}}}_2}_1}}\right)^{\sf{a_1}}}{\sf{a_1}}\pi}}[/tex]
[tex]\red{\large(\sf{^{n-2}e)\pi} }[/tex]
Então afirmar que o resultado a ser encontrado é [tex]\large(\sf{^{n-2}e)\pi} [/tex]
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