QuestionFor the following function, find the Taylor series centered at $x=\frac{\pi}{2}$ and then give the first 5 nonzero terms of the Taylor series and the open interval of convergence. $$f(x)=\cos (x)$$ $$f(x)=\sum_{n=0}^{\infty}$$ $\square$ $\square$ $\square$ $$+$$ $\square$ $$f(x)=\square 0_{0}+\square+\square a_{0}+\square$$ $\square$ $+\cdots$
The open interval of convergence is: (-infinity, infinity) $\square$ (Give your answer in interval notation.)
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Studdy Solution
Determine the interval of convergence:
The Taylor series for $\cos(x)$ centered at any point converges for all $x$. Therefore, the interval of convergence is:
$$(-\infty, \infty)$$
The first 5 nonzero terms of the Taylor series are:
$- (x-\frac{\pi}{2}) + \frac{1}{6}(x-\frac{\pi}{2})^3 - \frac{1}{120}(x-\frac{\pi}{2})^5 + \frac{1}{5040}(x-\frac{\pi}{2})^7 - \frac{1}{362880}(x-\frac{\pi}{2})^9$
The open interval of convergence is:
$$(-\infty, \infty)$$