March 2nd, 2016, 04:26 PM  #1 
Senior Member Joined: May 2012 Posts: 203 Thanks: 5  series expansion
I am supposed to show that : the sum of a(n)*Z^(2n) from n=0 to infinity is Maclaurin expansion of g(z)=f(Z^2) around 0. At first I thought I'd be differentiating using chain rule..but evaluating it at Z=0 doesn't get me very far.. Feel like I missing something obvious but I don't know what? 
March 2nd, 2016, 06:30 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,220 Thanks: 2411 Math Focus: Mainly analysis and algebra 
You are presumably given that $f(z) = \sum \limits_{n=0}^\infty a_nz^n$, from which we deduce that $f^{(n)}(0)$, the $n$th derivative of $f$ evaluated at $z=0$ is given by $f^{(n)}(0)=a_n$. Thus $g(z) = f(z^2)$ gives us $g(0)=f(0)=a_0$. And $g'(z) = 2zf'(z^2)$ so $g'(0)=0$. Then $g''(z) = 2f'(z^2) + 4z^2f'(z^2)$ and so $g''(0) = 2f'(0) = 2a_n$ as required. If you keep going down this path, you should get some insight into how the coefficients balance the factorials in the denominator of term of the Maclaurin expansion, and how all terms except the that containing the required derivative are zero. In particular, it seems that for odd derivatives of $g(z)$, each term will be multiplied by an odd power of $z$, with the highest derivative being the only one multiplied by just $z$. This term will produce the only term of the next (even) derivative of $g$ that is not multiplied by a positive power of $z$ and will thus not vanish at $z=0$. All you have to worry about are how the coefficients balance. 
March 7th, 2016, 11:02 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,966 Thanks: 807 
Given that $\displaystyle f(z)= \sum a_n z^n$, it follows, simply by replacing z with $\displaystyle z^2$, that $\displaystyle f(z^2)= \sum a_n (z^2)^n= \sum a_n z^{2n}$.

March 8th, 2016, 04:33 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,220 Thanks: 2411 Math Focus: Mainly analysis and algebra 
Certainly, but I thought that sounded a bit to straightforward as an answer to the OP. 

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