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Question Number 2157 by Yozzi last updated on 05/Nov/15

Evaluate 1+(2^3 /(1!))+(3^3 /(2!))+(4^3 /(3!))+... by considering the series expansion of an expression of the form P(x)e^x where P(x) is a suitably chosen polynomial in x.

$${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{1}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{3}!}+... \\ $$$${by}\:{considering}\:{the}\:{series}\:{expansion} \\ $$$${of}\:{an}\:{expression}\:{of}\:{the}\:{form}\:{P}\left({x}\right){e}^{{x}} \\ $$$${where}\:{P}\left({x}\right)\:{is}\:{a}\:{suitably}\:{chosen} \\ $$$${polynomial}\:{in}\:{x}.\: \\ $$$$ \\ $$$$ \\ $$

Commented by RasheedAhmad last updated on 15/Nov/15

1+(2^3 /(1!))+(3^3 /(2!))+(4^3 /(3!))+... =(1^3 /(0!))+(2^3 /(1!))+(3^3 /(2!))+(4^3 /(3!))+... =((1^3 .1)/(0!.1))+((2^3 .2)/(1!.2))+((3^3 .3)/(2!.3))+((4^3 .4)/(3!.4))+... =(1^4 /(1!))+(2^4 /(2!))+(3^4 /(3!))+(4^4 /(4!))+...+(n^4 /(n!))+...

$$\mathrm{1}+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{1}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{3}!}+... \\ $$$$=\frac{\mathrm{1}^{\mathrm{3}} }{\mathrm{0}!}+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{1}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{3}!}+... \\ $$$$=\frac{\mathrm{1}^{\mathrm{3}} .\mathrm{1}}{\mathrm{0}!.\mathrm{1}}+\frac{\mathrm{2}^{\mathrm{3}} .\mathrm{2}}{\mathrm{1}!.\mathrm{2}}+\frac{\mathrm{3}^{\mathrm{3}} .\mathrm{3}}{\mathrm{2}!.\mathrm{3}}+\frac{\mathrm{4}^{\mathrm{3}} .\mathrm{4}}{\mathrm{3}!.\mathrm{4}}+... \\ $$$$=\frac{\mathrm{1}^{\mathrm{4}} }{\mathrm{1}!}+\frac{\mathrm{2}^{\mathrm{4}} }{\mathrm{2}!}+\frac{\mathrm{3}^{\mathrm{4}} }{\mathrm{3}!}+\frac{\mathrm{4}^{\mathrm{4}} }{\mathrm{4}!}+...+\frac{{n}^{\mathrm{4}} }{{n}!}+... \\ $$$$ \\ $$

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