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September 23rd, 2015, 12:04 AM  #1 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  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! 
September 23rd, 2015, 12:24 AM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
The binomial expansion can be extended to realvalued exponents. If $r$ is a real number, then $(a + b)^r = a^r + ra^{r  1}b + \dfrac{r(r  1)}{2!}a^{r  2}b^2 + \dfrac{r(r  1)(r  2)}{3!}a^{r  3}b^3 + ...$ 
September 23rd, 2015, 06:38 AM  #3 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  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 workaround using the binomial series now, but thanks anyway 

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