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

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Answer 1 · 2015-12-17T00:05:23

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

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

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

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