# How to find the Maclurin Series with using the procedure method for $f(x)=\sqrt{1+2x}$

I stack about the following question

Use the procedure method to find the Maclaurin Series for $f(x)= \sqrt{1+2x}$

Also I would like to know what the procedure method is because I couldn't find the method in my text book.... So If you know the method, could you explain about it ?

Thanks !!

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There could be many "procedure" methods. In fact, any method is a procedure. – David Mitra Apr 19 '12 at 0:25

1. Use that for $a \in \Bbb R$${\left( {1 + x} \right)^a} = \sum\limits_{n = 0}^\infty {a \choose n}{x^n}$$ This will yield $$\sqrt{1 + 2x}= \sum\limits_{n = 0}^\infty {1/2 \choose n}2^n{x^n}$$ Now we need to find a closed form of a fractional binomial coefficient. This can be done rather "empirically", and I hope you also arrive at $${1/2 \choose n}= {\left( { - 1} \right)^{n + 1}}\frac{{(2n-3)\cdots5\cdot3\cdot1}}{{2^n\cdot n!}}$$ which gives you each coefficient of the expansion. 1. Another procedure would be computing each derivative at$x=0$and trying to find a pattern. If you do things right, you will arrive at $$f^{(n)}(0)= {\left( { - 1} \right)^{n + 1}}\left[ {\left( {2n - 3} \right) \cdots 5 \cdot 3 \cdot 1} \right]$$ i.e.$\left\{ f^{(n)}(0) \right\}=\{1,1,-1,3,-3\cdot 5,3\cdot5\cdot7,\dots\}$- [Edited in accord with the comments] Sorry, I don't know what "the procedure method" is. I do know one method that wroks quite nicely for this problem - use the binomial theorem which says that for all real$n$, $$(1+Q)^n=\sum_{k=0}^{\infty}{n\choose k}Q^k$$ - Shouldn't the limit of the sum be$n$, not$\infty$? – Javier Apr 19 '12 at 0:27 @JavierBadia Not really. Gerry is given the generalized binomial theroem, where$n\in \Bbb R$. I guess using$n$is confusing here. – Pedro Tamaroff Apr 19 '12 at 0:34 Indeed, in the problem at hand,$n\$ won't be an integer, and the series must be infinite. – Gerry Myerson Apr 19 '12 at 2:06