Mathematics I (Calculus) - Old Questions

3. (a) Find the Maclaurin series for cos x and prove that it represents cos x for all x.

We need to find derivatives of f(x) = cos x, so

Therefore, Maclaurin series for cos x  is

Since the cosine function and all the derivatives of cosine function have absolute value less than or equal to 1. So, by Taylor’s inequality

Now,

i.e.

 for all values of x.

This implies that the series converges to cosx for every value of x.