Menu Close

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-




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

Leave a Reply

Your email address will not be published. Required fields are marked *