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
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