- #1
MathematicalPhysicist
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i need to prove that:
1+x+x^2/2!+...+x^n/n!<=e^x<=1+x+x^2/2!+...+x^n/n!+(e^x)x^(n+1)/(n+1)! for x>=0
without using the power sum of e^x.
the textbook hints that i should evaluate the integral [tex]\int_{0}^{x}e^udu[/tex] and then i should integrate over and over n times,
and obtain the upper and lower limits for this integral.
i did the first part for the lower limit and i got that:
1+x+x^2/2+...+x^n/n<=e^x
the lhs of the last inequality is ofcourse bigger or equal than 1+x+x^2/2!+...+x^n/n!
my problem is with the upper bound of this inequality.
how to obtain it without using the power sum?
1+x+x^2/2!+...+x^n/n!<=e^x<=1+x+x^2/2!+...+x^n/n!+(e^x)x^(n+1)/(n+1)! for x>=0
without using the power sum of e^x.
the textbook hints that i should evaluate the integral [tex]\int_{0}^{x}e^udu[/tex] and then i should integrate over and over n times,
and obtain the upper and lower limits for this integral.
i did the first part for the lower limit and i got that:
1+x+x^2/2+...+x^n/n<=e^x
the lhs of the last inequality is ofcourse bigger or equal than 1+x+x^2/2!+...+x^n/n!
my problem is with the upper bound of this inequality.
how to obtain it without using the power sum?