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Calculus/Taylor Series sin(3x)

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Expert: - 3/30/2008


Question
I have to write a taylor series for f(x)=sin(3x) where c=pi/3

What is the series, interval, and radius of convergence?

Answer
The Taylor's Series is f(n)(x0)*f(x0)/n!.
f(n)(x0) is the nth derivative of f(x) evaluated at x0).

Here, I believe x0 is to be taken as pi/3 so the Taylor's Series will be positive for all terms.  In the Taylor's Series centered about 0, only the odd terms are positive and the rest are 0.  In evaluating
f(n)(x), note that a 3^n is gotten.

Convergence can be seen in http://en.wikipedia.org/wiki/Taylor_series

It will converge for all x, but if they are very far away from x0, it will take quite awhile.  The interval is really all x, but it sure takes a while if they are very far away.

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