obtain the Maclaurin series for the given function

Question

f(x) = xe^(4x)

Answer 1

Recall the Maclaurin expansion of e^x

e^x = 1 + x + x² / 2! + x³ / 3! + •••

which is valid for all x. Replacing x with 4x in the above equation yields

e^4x = 1 + 4x + (4x)² / 2! + (4x)³ + •••

Multiplying both sides by x yields Maclaurin expansion

xe^4x = x + 4x² + (4² / 2!)x³ + (4³ / 3!)x⁴ + •••