Expert: Ahmed Salami - 12/23/2009

Question
Hi Ahmed

My question relates to the Taylor series expansion of the binomial series (1 + x)^k.  (If you're unsure what I mean, please see: http://www.math10.com/en/university-math/taylor-series/taylor-series.html)  

In my unit guide, a Taylor series representation of (1 + x^2)^(-0.5) is obtained by simpling swapping x^2 for x in the Taylor series expansion of (1 + x)^k.  

Is this correct?  If x^2 were to appear instead of x in (1 + x)^k, wouldn't all subsequent derivatives of (1 + x^2)^k need to be worked by chain rule so that a repeating factor of 2x appears (making a straight swap of x^2 for x in the Taylor series expansion invalid)?


Thanks

Don


Answer
Hi Don,
It is in fact correct swapping x² for x in the Taylor Series expansion.
Now, its really good of you to have that kind of suspicion but you need to realise that the Taylor series expansion of a function is sort of an identity i.e the expressions on both sides of the equality sign are always the same (for every value where we have convergence). In the expansion of (1 + x)^(-½) you could replace x by anything and the resulting series would still be valid as long as we're in the limit of convergence.
If we tried to write the series for (1 + x²)^(-½), you would realise that the odd numbered derivatives would be zero and the series would only have even powers of x as expected. I think you should take the pain and try to find the first 3 or 4 terms of the expansion using the swapping and then obtaining you own Taylor series from scratch and see if it is different. It makes for better learning and understanding.
And of course you can always get back to me.

Regards