I am trying to work with Taylor series. What I am trying to do is to express $$ f(h) - f'(0) = f(h) - f'(h - h) $$ such that only derivatives of $f$ with the argument $h$ appear, for example $$ f(h) - f'(0) = f(h) - f'(h) + hf''(h) + \mathcal O(h^2) $$
I tried to do is this way:
f[h] - Series[f'[s - h], {h, 0, 2}] /. s -> h
Strangely, this gives me $$ \bigl(f(0) - f'(h)\bigr) - \bigl(f'(0) - f''(h)\bigr)h +\left(\frac{f''(0)}2 - \frac{f^{(3)}(h)}2\right)h^2 + \mathcal O(h^2) $$ which is not what I wanted. How can this behaviour be explained?
For the record, the correct answer can be obtained by
f[t] - Series[f'[s - h], {h, 0, 2}] /. {t -> h, s -> h}
f[h] - Series[f[h + δ], {δ, 0, 2}]
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