Question

How to write the first four terms of the Maclaurin series for the function f(x)=(x+1)e^(2x) given that ?

Answer 1

The first four terms of the Maclaurin series for `f(x)` are

`1 + 3x + 4x^2 + 10/3x^3`

Explanation:

The general form of a Maclaurin series is

`f(x) = sum_(n=0)^oo f^((n))(0)/(n!)x^n`

Then, the first four terms will be

`f(0)/(0!)x^0 + (f'(0))/(1!)x^1 + (f''(0))/(2!)x^2 + f^((3))(0)/(3!)x^3`

`= f(0) + f'(0)x + (f''(0))/2x^2 + f^((3))(0)/6x^3`

From the given function, we have

`f(0) = (0+1)e^(2*0) = 1`

`f'(0) = (2*0 + 3)e^(2*0) = 3`

`f''(0) = (4*0 + 8)e^(2*0) = 8`

`f^((3))(0) = (8*0+20)e^(2*0) = 20`

Thus the desired terms are

`1 + 3x + 4x^2 + 10/3x^3`