I'm interested in how would one find Taylor series of a function

$$f(z)=\left(1-\frac1z \right)^3$$ around $z_0=2$.

I have no clue where to start, the $3$ in the exponent baffles me. I tried to use the cube of the sum formula, but to no avail. I tried to find the series of a function $f(z)=1-1/z$ but I'm not sure where to go from there.

Any help would be appreciated.

  • $\begingroup$ If you know that the expansion of $1 - 1/z$ is, say, $\sum a_{i}z^{i}$ (as a Maclaurin expansion), simply take its third power. $\endgroup$ – Bill Wallis Apr 30 '18 at 17:57
  • $\begingroup$ @BillWallis I don't really know how to do that... $\endgroup$ – windircurse Apr 30 '18 at 17:58
  • $\begingroup$ Wouldn't it simply be $(1-\frac{1}{z})(1-\frac{1}{z})(1-\frac{1}{z})=1-\frac{2}{z}+\frac{2}{z^2}-\frac{1}{z^3}$ $\endgroup$ – Rhys Hughes Apr 30 '18 at 18:02
  • 2
    $\begingroup$ @RhysHughes That is not a Taylor series, which must have the form $\sum c_i(x-a)^i$. $\endgroup$ – MJD Apr 30 '18 at 18:03
  • $\begingroup$ @MJD. Gotcha, thanks $\endgroup$ – Rhys Hughes Apr 30 '18 at 18:42

Note that$$\left(1-\frac1z\right)^3=1-\frac3z+\frac3{z^2}-\frac1{z^3}.\tag1$$Now,\begin{align}\frac1z&=\frac1{2+(z-2)}\\&=\frac12\cdot\frac1{1+\frac{z-2}2}\\&=\frac12-\frac{z-2}{2^2}+\frac{(z-2)^2}{2^3}-\cdots\end{align}if $|z-2|<2$. Therefore\begin{align}\frac1{z^2}&=-\left(\frac1z\right)'\\&=\frac1{2^2}-\frac{2(z-2)}{2^3}+\frac{3(z-2)^2}{2^4}-\cdots\end{align}and\begin{align}\frac1{z^3}&=-\frac12\left(\frac1{z^2}\right)'\\&=\frac{2}{2^4}-\frac{2\times3(z-2)}{2^5}+\frac{3\times4(z-2)^2}{2^6}-\cdots\end{align}Now, use $(1)$ to get everything as a single series.

  • First: it's easier around $0$ (as you can use "standard results" and compose known Taylor expansions). So write $z=2+x$, from which $$ f(z) = \left(1-\frac{1}{2+x}\right)^3 \tag{1} $$ which you now want to expand around $x_0=0$.

  • Once you have (1), $$ \left(1-\frac{1}{2+x}\right)^3 = \frac{1}{2^3}\left(2-\frac{1}{1+x/2}\right)^3 = \frac{1}{2^3}\left(2-\left(1-\frac{x}{2}+o(x)\right)\right)^3\tag{2} $$ using the series of $\frac{1}{1+u}$ for $u$ around $0$. So you get $$ \left(1-\frac{1}{2+x}\right)^3 = \frac{1}{2^3}\left(1+\frac{x}{2}+o(x)\right)^3 = \frac{1}{2^3}\left(1+\frac{3x}{2}+o(x)\right) = \frac{1}{8}+\frac{3x}{16}+o(x) \tag{3} $$ using the series of $(1+u)^3$ for $u$ around $0$.

  • Finally, you can "get back to $z$" from (3): $$ f(z) = \frac{1}{8} + \frac{3}{16}(z-2)+ o(z-2) $$

Of course, the above works as well if you want to expand to a higher order. Just do the same order for $x$ around $0$.


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