Find the maclaurin series of the function f (x) = (3x2) e-7xf (x) = (3x2) e-7x (f (x) = ∑n=0∞cnxn) (f (x) = ∑n=0∞cnxn)

If you know the series for e^x, then you can start from there:

e^x = sum (n from 0 to inf) [ (x^n) / n!]

=> e^3x = sum (n from 0 to inf) [ (3x) ^n / n!]

If you don't know the series for e^x, then do it this way:

f (x) = f (0) + x f' (0) + x^2 f'' (0) / 2! + x^3 f''' (0) / 3! + ...

f (0) = e^0 = 1

f' (x) = 3e^3x = > f' (0) = 3

f'' (x) = (3^2) e^3x = > f' (0) = 3^2

f''' (x) = (3^3) e^3x = > f' (0) = 3^3

Re-arranging and you'll get the sigma form as above.