this is my first time posting so hopefully I'm not messing things up.
I understand the general idea behind Taylor series, but the I don't understand what's meant by "We may repeat this procedure" in this book (Riley's Mathematical methods). I've written in green what I think I understand, and the part in red is the source of my problem.
It looks to me that the procedure is substituting derivatives of $f(x)$ into $f(a+h)$, but $f(a+h)$ only contains $f'(x)$, and no higher order derivatives? So how am I supposed to "repeat the procedure"?
I know there are other ways of achieving the end result, but I it is this particular method I'm interested in learning.
Thanks in advance
The above line of your book perhaps means we'd keep doing integration by parts again and again and will obtain a series which is Taylor Series .
Integration by Parts:
$$ \int u(x)v'(x) \, dx = u(x)v(x) - \int v(x) u'(x) \, dx $$
Do, that to the equation which is just before (algebra) denoted in green, you'll get Taylor Series.