Please help me with this problem?

Question

Which of the following is the coefficient of x^4 in the Maclaurin series generated by cos (3x)? 

 

a) 27/8

b) 9

c) 1/24

d) 0

e) -27/8

 

Please help me step by step. Show all your work. 

Answer 1

You know that f(x) can be expanded as a series as follows. 

 

f(x) = f(0) + f'(0)*x + (f''(0)/2)*x² + (f'''(0)/6)*x³ + (f''''(0)/24)*x⁴ + ...

 

In case of cos(x), the coefficient of any odd power is zero, which matches well with the fact that cos(x) and even power are even functions.

 

cos(x) = 1 -1/2*x² + 1/24*x⁴ + ...

 

When we apply the chain rule in computing derivatives of cos(3x), the Maclaurin series for cos(3x) is computed as follows,

 

cos(3x) = 1 -(3²/2)*x² + (3⁴/24)*x⁴ + ....

 

So, the coefficient of x⁴ is 3⁴/24, which is 27/8. 

 

The answer: a) 27/8