Binomial expansion question. easy!

Question

I'm trying to do the binomial expansion of $(x-2)^{1/2}$.

How do you do it? As far as I'm aware the expansion only works for $(1+x)^n$. How could I get it in that form?

Thanks.

Answer 1

One way to do this is to rewrite the expression as $(1+(x-3))^{1/2}$, and then substitute $u=x-3$ and expand $(1+u)^{1/2}$ using the Binomial Series $\displaystyle(1+u)^r=\sum_{n=0}^{\infty}\binom{r}{n}u^n$.

Answer 2

If what you meant to ask about was a binomial expansion, then the approach is as follows:

First, note that the binomial theorem states that \begin{align*} (a+b)^n = \sum_{k=0}^{\infty}\binom{n}{k}a^kb^{n-k} \end{align*} Thus, by substituting for the above values, we have \begin{align*} (x-2)^\frac{1}{2} = \sum_{k=0}^{\infty}\binom{\frac{1}{2}}{k}x^k(-2)^{\frac{1}{2}-k} \end{align*}

Answer 3

The basic binomial formula $(1+x)^\alpha=\sum_k\binom\alpha kx^k$ is valid as a formal power series in $x$, or for concrete numbers$~x$ with $|x|<1$ (supposing that like is the case here, $\alpha$ is not a natural number). This cannot be used directly in the example, but you could bring the expression into a similar form by extracting a factor $-2$ from the expression $x-2$, writing $(x-2)^\alpha=(-2(1-\frac x2))^\alpha=(-2)^\alpha(1-\frac x2)^\alpha$ for $\alpha=\frac12$.

But this brings to the front a question that one should have considered at the beginning: what is the value $(-2)^\frac12$ that your formula gives at $x=0$, and will have to be the constant term of any reasonable expansion? The value $\sqrt{-2}$ is not very well defined; it does not exist as real number, and as complex number it it could be either $\def\i{\mathbf i}\sqrt2\i$ or $-\sqrt2\i$. So a formula with real numbers valid near $x=0$ is just not possible; if you want a formula that describes one of the complex square roots of $x-2$ near $x=0$ you can choose $r$ to be one of the mentioned square roots of $-2$, and write $$(x-2)^\frac12=r\sqrt{1-\frac x2} =r\sum_k\binom{1/2}k\Bigl(-\frac1{2^k}\Bigr)x^k. $$ This series (which has purely imaginary coefficients due to the factor$~r$) converges for $|x|<2$.