Express the definite integral as an infinite series and find its value to within an error of at most 10−4 ∫01cos⁡(x2)dx

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Express the definite integral as an infinite series and find its value to within an error of at most  10−4  ∫01cos⁡(x2)dx

i1ziZ · Accepted Answer

Term-by-Term Differentiation and IntegrationF(x)=∑n=0∞an(x−c)nhas radius of convergence R&amp;gt;0. Then F is differentiable on (c−R,c+R). Furtehermore, we can integrate and differentiate term by term. For x∈(c−R,c+R)F′(x)=∑n=1∞nan(x−c)n−1∫F(x)dx=A+∑n=0∞ann+1(x−c)n+1 (A any constant)These series have the same radius of convergence RHere we need to find the value of F(x)=∫01cos⁡(x2)dx we will use the expansion of the cos⁡xFrom table 2, We have Maclaurin Series f(x)=cos⁡x=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!−x66!+… converges for for all x here x is replaced with x2cos⁡x2=∑n=0∞(−1)n(x2)2n(2n)!cos⁡x2=∑n=0∞(−1)nx4n(2n)!dx∫01cos⁡x2dx=∑n=0∈ft(−1)n1(4n+1)(2n)!Now we need to find out F(1) error less than 0.0001∫01cos⁡x2dx=∑n=0∞(−1)n1(4n+1)(2n)!Above is alternating series with

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

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