6.4 Working with taylor series  (Page 2/11)

We now need to determine the interval of convergence for the binomial series [link] . We apply the ratio test. Consequently, we consider

Since

if and only if $\left|x\right|<1,$ we conclude that the interval of convergence for the binomial series is $\left(\mathrm{-1},1\right).$ The behavior at the endpoints depends on $r.$ It can be shown that for $r\ge 0$ the series converges at both endpoints; for $\mathrm{-1}<r<0,$ the series converges at $x=1$ and diverges at $x=\mathrm{-1};$ and for $r<\mathrm{-1},$ the series diverges at both endpoints. The binomial series does converge to ${\left(1+x\right)}^r$ in $\left(\mathrm{-1},1\right)$ for all real numbers $r,$ but proving this fact by showing that the remainder $R_n\left(x\right)\to 0$ is difficult.

We can use this definition to find the binomial series for $f\left(x\right)=\sqrt{1+x}$ and use the series to approximate $\sqrt{1.5}.$

Common functions expressed as taylor series

At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form $f\left(x\right)={\left(1+x\right)}^r.$ In [link] , we summarize the results of these series. We remark that the convergence of the Maclaurin series for $f\left(x\right)=\text{ln}\left(1+x\right)$ at the endpoint $x=1$ and the Maclaurin series for $f\left(x\right)={\text{tan}}^{\mathrm{-1}}x$ at the endpoints $x=1$ and $x=\mathrm{-1}$ relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.)

Earlier in the chapter, we showed how you could combine power series to create new power series. Here we use these properties, combined with the Maclaurin series in [link] , to create Maclaurin series for other functions.