Chapter 12

Set \(f(x)=\ln (1+x) .\) Using a graphing utility or a CAS. draw a figure that gives the graph of \(f\) and the graphs of the Taylor polynomials \(P_{2}, P_{3}, P_{4}, P_{3}\).

Show that \(e\) is irrational by taking the following steps. (1) Begin with the expansion $$e=\sum_{k=0}^{\infty} \frac{1}{k !}$$ and show that the \(q\) th partial sum $$s_{q}=\sum_{k=0}^{\infty} \frac{1}{k !}$$ satisfies the inequality $$0