Hello

I hope you are doing good.

The expansion of sin(x) is -

Let f ( x ) = sin( x ). To find the Maclaurin series coefficients, we must evaluate

for k = 0, 1, 2, 3, 4, ...

Calculating the first few coefficients, a pattern emerges:

The coefficients alternate between 0, 1, and -1. You should be able to, for the n th derivative, determine whether the n th coefficient is 0, 1, or -1.

Substitute coefficients into expansion

Thus, the Maclaurin series for sin( x ) is

From the first few terms that we have calculated, we can see a pattern that allows us to derive an expansion for the n th term in the series, which is

Substituting this into the formula for the Taylor series expansion, we obtain

Regards

Prashant Singh