I wish to prove the following.
"Let be represented by Taylor series at , where , and let the first non-vanishing derivatives at be and , where . Then
"
So here's how I've proceeded so far:
If we let , then
Differentiating wrt :
Again, letting , then
And so on, so that:
and similarly,
So:
Recalling that and are the first non-vanishing derivatives at . Then:
Recalling that , then:
But unless , as , and so , which clearly is not the desired result.
Where am I going wrong?