Respuesta :
Answer:
a) [tex] n =0, \frac{(-1)^0}{2*0+1} x^{2*0+1}= x[/tex]
[tex] n =1, \frac{(-1)^1}{2*1+1} x^{2*1+1}= -\frac{x^3}{3}[/tex]
[tex] n =2, \frac{(-1)^2}{2*2+1} x^{2*2+1}= \frac{x^5}{5}[/tex]
[tex] n =3, \frac{(-1)^3}{2*3+1} x^{2*3+1}= -\frac{x^7}{7}[/tex]
b) n=0
[tex] arctan(\pi/6) \approx \pi/6 = 0.523599[/tex]
The real value for the expression is [tex] arctan (\pi/6) = 0.482348[/tex]
And if we replace into the formula of relative error we got:
[tex] \% error= \frac{|0.523599 -0.482348|}{0.482348} * 100= 8.55\%[/tex]
n =1
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} = 0.47576[/tex]
[tex] \% error= \frac{|0.47576 -0.482348|}{0.482348} * 100= 1.37\%[/tex]
n =2
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} +\frac{(pi/6)^5}{5} = 0.483631[/tex]
[tex] \% error= \frac{|0.483631 -0.482348|}{0.482348} * 100= 0.27\%[/tex]
n =3
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} +\frac{(pi/6)^5}{5}-\frac{(pi/6)^7}{7} = 0.48209[/tex]
[tex] \% error= \frac{|0.48209 -0.482348|}{0.482348} * 100= 0.05\%[/tex]
[tex] \arctan (\pi/6) = 0.48[/tex]
Step-by-step explanation:
Part a
the general term is given by:
[tex] a_n = \frac{(-1)^n}{2n+1} x^{2n+1}[/tex]
And if we replace n=0,1,2,3 we have the first four terms like this:
[tex] n =0, \frac{(-1)^0}{2*0+1} x^{2*0+1}= x[/tex]
[tex] n =1, \frac{(-1)^1}{2*1+1} x^{2*1+1}= -\frac{x^3}{3}[/tex]
[tex] n =2, \frac{(-1)^2}{2*2+1} x^{2*2+1}= \frac{x^5}{5}[/tex]
[tex] n =3, \frac{(-1)^3}{2*3+1} x^{2*3+1}= -\frac{x^7}{7}[/tex]
Part b
If we use the approximation [tex] arctan x \approx x[/tex] we got:
n=0
[tex] arctan(\pi/6) \approx \pi/6 = 0.523599[/tex]
The real value for the expression is [tex] arctan (\pi/6) = 0.482348[/tex]
And if we replace into the formula of relative error we got:
[tex] \% error= \frac{|0.523599 -0.482348|}{0.482348} * 100= 8.55\%[/tex]
If we add the terms for each value of n and we calculate the error we see this:
n =1
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} = 0.47576[/tex]
[tex] \% error= \frac{|0.47576 -0.482348|}{0.482348} * 100= 1.37\%[/tex]
n =2
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} +\frac{(pi/6)^5}{5} = 0.483631[/tex]
[tex] \% error= \frac{|0.483631 -0.482348|}{0.482348} * 100= 0.27\%[/tex]
n =3
[tex] arctan(\pi/6) \approx \pi/6 -\frac{(pi/6)^3}{3} +\frac{(pi/6)^5}{5}-\frac{(pi/6)^7}{7} = 0.48209[/tex]
[tex] \% error= \frac{|0.48209 -0.482348|}{0.482348} * 100= 0.05\%[/tex]
And thn we can conclude that the approximation is given by:
[tex] \arctan (\pi/6) = 0.48[/tex]
Rounded to 2 significant figures
