Answer to Question #206423 in Calculus for Oula

Question #206423

Use the exponential series to find an approximation to e-1correct to three significant figures.

 


1
Expert's answer
2021-06-16T11:05:59-0400
ex=1+x1!+x22!+x33!+x44!+...e^x=1+\dfrac{x}{1!}+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+...

=n=0xnn!   for all x=\displaystyle\sum_{n=0}^{\infin}\dfrac{x^n}{n!} \ \ \text{ for all }x

e1=1+11!+(1)22!+(1)33!+(1)44!e^{-1}=1+\dfrac{-1}{1!}+\dfrac{(-1)^2}{2!}+\dfrac{(-1)^3}{3!}+\dfrac{(-1)^4}{4!}

+(1)55!+(1)66!+(1)77!+(1)88!+...+\dfrac{(-1)^5}{5!}+\dfrac{(-1)^6}{6!}+\dfrac{(-1)^7}{7!}+\dfrac{(-1)^8}{8!}+...

=11+1216+1241120+172015040=1-1+\dfrac{1}{2}-\dfrac{1}{6}+\dfrac{1}{24}-\dfrac{1}{120}+\dfrac{1}{720}-\dfrac{1}{5040}

+140320...+\dfrac{1}{40320}-...

1403200.0000245\dfrac{1}{40320}\approx0.0000245



201606720+1680336+561403201483940320\approx\dfrac{20160-6720+1680-336+56-1}{40320}\approx\dfrac{14839}{40320}


0.368\approx0.368

e1=0.368e^{-1}=0.368



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