Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions.

e^-5x

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EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos x = = 2 = 1 + x + 2 1 x + X 3.x 3 3! = (¹ + x + — + 1 x³ 3 + + 6 4! x5 3.4! x2k (2k)! + (-1). = 1 ¹ + x + £ + § + ¾ + · · · ) · ( ¹ - # - # ----) 1 2! 3! 4! 4! + 72 + + - - -) - ( ₁ ++ ++) 3! 4! 2!2! 2!3! + +6 214! کر +