91Ó°ÊÓ

Find the constant term of the Fourier series of the triangular wave function defined by \(f(x)=|x|\) for \(-1 \leq\) \(x \leq 1\) and \(f(x+2)=f(x)\) for all \(x\).

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\) $$\sqrt{1-x}, \quad a=0, \quad n=3$$

Using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\cosh t$$

the Taylor polynomial \(P_{n}(x)\) about 0 approximates \(f(x)\) with error \(E_{n}(x)\) and the Taylor series converges to \(f(x)\) on \(|x| \leq 1 .\) Find the smallest constant \(K\) given by the alternating series error bound such that \(\left|E_{6}(1)\right| \leq K\) $$\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{2^{n}}$$

For Exercises \(10-17\), find the first four terms of the Taylor series for the function about the point \(a\). $$\cos t, \quad a=\pi / 6$$

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\) $$e^{x}, \quad a=1, \quad n=4$$

Using known Taylor series, find the first four nonzero terms of the Taylor series about 0 for the function. $$\sinh t$$

Find the Taylor series about 0 for the function. Include the general term. $$(1+x)^{3}$$

Find the Taylor polynomial of degree \(n\) for \(x\) near the given point \(a\) $$\frac{1}{1+x}, \quad a=2, \quad n=4$$

For Exercises \(10-17\), find the first four terms of the Taylor series for the function about the point \(a\). $$\sin \theta, \quad a=-\pi / 4$$