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You are saying that you "change$w_2$ a little bit", but then the equation also updates $w_N$ , which is wrong at this point. (This is only done later.)
Description of changes: I am just removing$+\epsilon_N$ from the end of the equation $L(w_1+\epsilon_1, w_2+\epsilon_2, \ldots, w_N) \approx L(w_1, w_2+\epsilon_2, \ldots, w_N) + \epsilon_1 \frac{\partial}{\partial w_1} L(w_1, w_2+\epsilon_2, \ldots, w_N+\epsilon_N)$ so that it nou reads $L(w_1+\epsilon_1, w_2+\epsilon_2, \ldots, w_N) \approx L(w_1, w_2+\epsilon_2, \ldots, w_N) + \epsilon_1 \frac{\partial}{\partial w_1} L(w_1, w_2+\epsilon_2, \ldots, w_N)$ .
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