obtain the Maclaurin series for the given function help pls

Question

f(x) = 4x cos (1/9 * x^2)

Answer 1

Rather than use the function and calculate the Maclaurin Series directly, it is easier to plug in x²/9 for x in the series for cos(x) and, then, multiply each term by 4x:

cos(x) = 1-x²/2! + x⁴/4! - x⁶/6! + ... or summation from i = 0 to ∞ of (-1)ⁱx²ⁱ/(2i)!

cos(x²/9) = summation from i = 0 to ∞ of (-1)ⁱ(x⁴ⁱ/9²ⁱ)/(2i)!

4x*cos(x²/9) = summation from i = 0 to ∞ of 4(-1)ⁱ(x⁴ⁱ⁺¹/9²ⁱ)/(2i)!

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