Question

6. Follow the steps to find a power series representation of the function with center and determine the radius of convergence (x) =tan(x) a) Note that shox tan"'(x). So, start with the function (x) and determine a power series representation of it (using the sum of a geometric series b) What is the radius of convergence of A(x)? c) Integrate the power series you found in (a). This is the power series representation of B(x) = tan-'(x). d) Find the constant of integration)by setting x = 0 and solving e) What is the radius of convergence of B(x)? Rewrite the power series you found in (c).plugging in the value you found for C. g) Finally, replace x with x in the power series from () to get a power series representation of the original function f(x). h) What is the interval of convergence of f(x)?

Answer 1

A 6 stále (1+x2) - XⓇ+ -- +... l-am 4x4 Žanan an= =(1)/2 neven. nodd. (6) lim lan ² neven nodd. sup lim lan12=1 So, radius of convergent of Alnya, JA (2) - JC1 -&rtan -x6 ) da tc. = B(x) csays. tanin. (d) when, x=0, cao, tanals - + anan n=0 an O n even. ans nodd. n