Activity 8.5.4.
(a)
Plot the graphs of several of the Taylor polynomials centered at \(0\) (of order at least 5) for \(e^x\) and convince yourself that these Taylor polynomials converge to \(e^x\) for every value of \(x\text{.}\)
(b)
Draw the graphs of several of the Taylor polynomials centered at \(0\) (of order at least 6) for \(\cos(x)\) and convince yourself that these Taylor polynomials converge to \(\cos(x)\) for every value of \(x\text{.}\) Write the Taylor series centered at \(0\) for \(\cos(x)\text{.}\)
(c)
Draw the graphs of several of the Taylor polynomials centered at \(0\) for \(\frac{1}{1-x}\text{.}\) Based on your graphs, for what values of \(x\) do these Taylor polynomials appear to converge to \(\frac{1}{1-x}\text{?}\) How is this situation different from what we observe with \(e^x\) and \(\cos(x)\text{?}\) In addition, write the Taylor series centered at \(0\) for \(\frac{1}{1-x}\text{.}\)
