4 General power series

A general power series has the form

$b_0+b_1\left(x-x_0\right)+b_2{\left(x-x_0\right)}^2+\cdots ={\sum }_{p=0}^{\infty }{b}_p{\left(x-x_0\right)}^p$

Exactly the same considerations apply to this general power series as apply to the ‘special’ series ${\sum }_{p=0}^{\infty }{b}_p{x}^p$ except that the variable $x$ is replaced by $\left(x-x_0\right)$ . The radius of convergence of the general series is obtained in the same way:

$R=\underset{p\to \infty }{lim}\left|\frac{b_p}{b_{p+1}}\right|$

and the interval of convergence is now shifted to have centre at $x=x_0$ (see Figure 4 below). The series is absolutely convergent if $\left|x-x_0\right|<R$ , diverges if $|x-x_0|>R$ and may or may not converge if $|x-x_0|=R.$

Figure 4

Task!

Find the radius of convergence of the general power series

$1-\left(x-1\right)+{\left(x-1\right)}^2-{\left(x-1\right)}^3+\cdots$

First find an expression for the general term:

Answer

Now obtain the radius of convergence:

Answer

Finally, decide on the convergence at $\left|x-1\right|=1$ (i.e. at $x-1=-1$ and $x-1=1$ i.e. $x=0$ and $x=2$ ):

Answer

Exercises

1. From the result $\frac{1}{1-x}=1+x+x^2+x^3+\dots ,\left|x\right|<1$
   1. Find an expression for $ln\left(1-x\right)$
   2. Use this expression to obtain an approximation to $ln\left(0.9\right)$ to 4 d.p.
2. Find the radius of convergence of the general power series  $1-\left(x+2\right)+{\left(x+2\right)}^2-{\left(x+2\right)}^3+\dots$
3. Find the range of values of $x$ for which the power series $1+\frac{x}{4}+\frac{x^2}{4^2}+\frac{x^3}{4^3}+\dots$ converges.
4. By differentiating the series for ${\left(1+x\right)}^{1∕3}$ find the power series for ${\left(1+x\right)}^{-2∕3}$ and state its radius of convergence.
5. 1. Find the radius of convergence of the series $1+\frac{x}{3}+\frac{x^2}{4}+\frac{x^3}{5}+\dots$
   2. Investigate what happens at the points $x=-1$ and $x=+1$

Answer