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September 23rd, 2015, 12:04 AM   #1
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Binomial series question

In the theory associated with the magnetic field due to an electric current, the expression $\displaystyle 1 - \frac{x}{\sqrt{a^2 +x^2}}$ is found. By expanding $\displaystyle (a^2 + x^2)^{-1/2}$, find the first three nonzero terms that could be used to approximate the given expression.

The binomial theorem doesn't seem to work as the power is negative and a fraction. However, the expression inside the round brackets isn't in the form of $\displaystyle 1 + x$, so how do I go about manipulating it to use the binomial series? Thanks!
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September 23rd, 2015, 12:24 AM   #2
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The binomial expansion can be extended to real-valued exponents.

If $r$ is a real number, then $(a + b)^r = a^r + ra^{r - 1}b + \dfrac{r(r - 1)}{2!}a^{r - 2}b^2 + \dfrac{r(r - 1)(r - 2)}{3!}a^{r - 3}b^3 + ...$
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September 23rd, 2015, 06:38 AM   #3
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Quote:
Originally Posted by Azzajazz View Post
The binomial expansion can be extended to real-valued exponents.

If $r$ is a real number, then $(a + b)^r = a^r + ra^{r - 1}b + \dfrac{r(r - 1)}{2!}a^{r - 2}b^2 + \dfrac{r(r - 1)(r - 2)}{3!}a^{r - 3}b^3 + ...$
Thanks for the info. We weren't taught to use the binomial theorem for negative and fractional exponents, so I don't think I'll be allowed to use this for long questions. I've found a work-around using the binomial series now, but thanks anyway
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