Find the Maclaurin series for the function. Use the table of power series for elementary functions shown below.

f(x) = sin 5x

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POWER SERIES FOR ELEMENTARY FUNCTIONS Function X 1 (x-1) + (x − 1)²(x - 1)³ + (x - 1)4+ (-1)^(x - 1)" + 1 + x = 1- x + x²-x³ + ₁²-1³ + . .. In x = (x - 1) et = 1 + x sin x = x - cos x = 1 + arctan xx 1³ 3! 2! + arcsin x = x + (x-1)2(x-1)³ (x - 1)4 + + ps 5! + 2 T 7! 4! 6! + + + 5! 9! 8! x² - ² + ²/² - / +²² 3 5 7 3. 13 1.3x³ 2.3 2.4.5 + + + + (-1)" x +.. (-1)"x²n+1 (2n + 1)! + +...+ (-1)" x²n (2n)! 1.3.5x7 2.4.6.7 (-1)²x²m+1 2n + 1 + k(k-1)(k 2)x³ (1+x) = 1 + kx + k(k-1)x² 2! 3! * The convergence at X = ±1 depends on the value of k. +... +. + (-1)²-¹(x-1)" n (2n)!x²n+1 (2^n!)²(2n + 1) +. + k(k-1). (k n! + .. n+1)x Interval of Convergence 0 < x < 2 -1 < x < 1 0 < x≤ 2 -∞0 < x < ∞ -∞0 < x < ∞ -∞0 < x < 0 -1 ≤ x ≤ 1 -1 ≤ x ≤ 1 -1 < x < 1*