91影视

Suppose that \(p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) such that \(a_{n}=0\) if \(n\) is even. Explain why \(p(x)=-p(-x)\).

In the following exercises, find the radius of convergence of the Maclaurin series of each function. $$ \ln (1+x) $$

In the following exercises, given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\), use term-by-term differentiation or integration to find power series for each function centered at the given point. \(f(x)=\frac{2 x}{\left(1-x^{2}\right)^{2}}\) at \(x=0\)

In the following exercises, find the radius of convergence of the Maclaurin series of each function. $$ \frac{1}{1+x^{2}} $$

Suppose that \(p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) such that \(a_{n}=0\) if \(n\) is odd. Explain why \(p(x)=p(-x)\).

In the following exercises, given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\), use term-by-term differentiation or integration to find power series for each function centered at the given point. \(f(x)=\tan ^{-1}\left(x^{2}\right)\) at \(x=0\)

The following exercises make use of the functions \(S_{5}(x)=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\) and \(C_{4}(x)=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}\) on \([-\pi, \pi]\). [T] Plot \(e^{x}-e_{4}(x)\) where \(e_{4}(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) on \([0,2] .\) Compare the maximum error with the Taylor remainder estimate.

In the following exercises, find the radius of convergence of the Maclaurin series of each function. $$ \tan ^{-1} x $$

In the following exercises, given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\), use term-by-term differentiation or integration to find power series for each function centered at the given point. \(f(x)=\ln \left(1+x^{2}\right)\) at \(x=0\)

Suppose that \(p(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) converges on \((-1,1] .\) Find the interval of convergence of \(p(2 x-1)\).