Express the definite integral as an infinite series and find its value to within

Globokim8 2021-11-05 Answered
Express the definite integral as an infinite series and find its value to within an error of at most
104
01cos(x2)dx
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Expert Answer

i1ziZ
Answered 2021-11-06 Author has 92 answers
Term-by-Term Differentiation and Integration
F(x)=n=0an(xc)n
has radius of convergence R>0. Then F is differentiable on (cR,c+R). Furtehermore, we can integrate and differentiate term by term. For x(cR,c+R)
F(x)=n=1nan(xc)n1
F(x)dx=A+n=0ann+1(xc)n+1 (A any constant)
These series have the same radius of convergence R
Here we need to find the value of F(x)=01cos(x2)dx we will use the expansion of the cosx
From table 2, We have Maclaurin Series f(x)=cosx=n=0(1)nx2n(2n)!=1x22!+x44!x66!+ converges for for all x here x is replaced with x2
cosx2=n=0(1)n(x2)2n(2n)!
cosx2=n=0(1)nx4n(2n)!dx
01cosx2dx=n=0ft(1)n1(4n+1)(2n)!
Now we need to find out F(1) error less than 0.0001
01cosx2dx=n=0(1)n1(4n+1)(2n)!
Above is alternating series with

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