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Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=t7+1, y=t5+t; t=71 y: Need Help? _ Find the area of the region enclosed by one loop of the curve. r = sin(12n9) EXAMPLE 7 Find a power series representation for f(x) = arctan(x). SOLUTION We observe that f '(x) = 1 / (1 + x ) and find the required series by integrating the power series for 1 / (1 + x-). arctan(x) = J dx (1 + x2 ) = 1(1 - + x 4 + ...)dx =C+x- + + To find C, we put x = 0 and obtain C = arctan(0) = 0. Therefore = X + +... UT = E (-1)7 n=0 Since the radius of convergence of the series for 1/(1 + x ) is , the radius of convergence of this series is alsoEXAMPLE 5 Express 110. 7 X}2 as a power series by differentiating the equation below. What is the radius of convergence? 1 2 m =l+x+x+x3+-n= x